Financial products having demand-based, adjustable returns, and trading exchange therefor

ABSTRACT

This invention provides methods and systems for trading and investing in groups of demand-based adjustable-return contingent claims, and for establishing markets and exchanges for such claims. (FIG.  2,  item  262, 263, 264, 265 ) The advantages of the present invention, as applied to the derivative securities and similar financial markets, include increased liquidity, reduced credit risk, improved information aggregation, increased price transparency, reduced settlement or clearing costs, reduced hedging costs, reduced model risk, reduced event risk, increased liquidity incentives, improved self-consistency, reduced influence by market makers, and increased ability to generate and replicate arbitrary payout distributions. In addition to the trading of derivative securities, the present invention also facilitates the trading of other financial-related contingent claims; non-financial-related contingent claims such as energy, commodity, and weather derivatives; traditional insurance and reinsurance contracts; and contingent claims relating to events which have generally not been readily insurable or hedgeable such as corporate earnings announcements, future semiconductor demand, and changes in technology.

This application represents a national filing under 35 U.S.C. § 371 ofInternational Application No. PCT/US00/19447, filed Jul. 18, 2000, whichclaims priority to U.S. Provisional App. No. 60/144,890, filed Jul. 21,1999, and U.S. patent application Ser. No. 09/448,822, filed Nov. 24,1999, now U.S. Pat. No. 6,321,212 B1.

COPYRIGHT NOTICE

This document contains material which is subject to copyrightprotection. The applicant has no objection to the facsimile reproductionof this patent document, as it appears in the U.S. Patent and TrademarkOffice (PTO) patent file or records or in any publication by the PTO orcounterpart foreign or international instrumentalities. The applicantotherwise reserves all copyright rights whatsoever.

FIELD OF THE INVENTION

This invention relates to systems and methods for demand-based trading.More specifically, this invention relates to methods and systems fortrading financial products having demand-based adjustable returns, andsystems and methods for determining those returns.

BACKGROUND OF THE INVENTION

With the rapid increase in usage and popularity of the public Internet,the growth of electronic Internet-based trading of securities has beendramatic. In the first part of 1999, online trading via the Internet wasestimated to make up approximately 15% of all stock trades. This volumehas been growing at an annual rate of approximately 50%. High growthrates are projected to continue for the next few years, as increasingvolumes of Internet users use online trading accounts.

Online trading firms such as E-Trade Group, Charles Schwab, andAmeritrade have all experienced significant growth in revenues due toincreases in online trading activity. These companies currently offerInternet-based stock trading services, which provide greater convenienceand lower commission rates for many retail investors, compared totraditional securities brokerage services. Many expect online trading toexpand to financial products other than equities, such as bonds, foreignexchange, and financial instrument derivatives.

Financial products such as stocks, bonds, foreign exchange contracts,exchange traded futures and options, as well as contractual assets orliabilities such as reinsurance contracts or interest-rate swaps, allinvolve some measure of risk. The risks inherent in such products are afunction of many factors, including the uncertainty of events, such asthe Federal Reserve's determination to increase the discount rate, asudden increase in commodity prices, the change in value of anunderlying index such as the Dow Jones Industrial Average, or an overallincrease in investor risk aversion. In order to better analyze thenature of such risks, financial economists often treat the real-worldfinancial products as if they were combinations of simpler, hypotheticalfinancial products. These hypothetical financial products typically aredesigned to pay one unit of currency, say one dollar, to the trader orinvestor if a particular outcome among a set of possible outcomesoccurs. Possible outcomes may be said to fall within “states,” which aretypically constructed from a distribution of possible outcomes (e.g.,the magnitude of the change in the Federal Reserve discount rate) owingto some real-world event (e.g., a decision of the Federal Reserveregarding the discount rate). In such hypothetical financial products, aset of states is typically chosen so that the states are mutuallyexclusive and the set collectively covers or exhausts all possibleoutcomes for the event. This arrangement entails that, by design,exactly one state always occurs based on the event outcome.

These hypothetical financial products (also known as Arrow-Debreusecurities, state securities, or pure securities) are designed toisolate and break-down complex risks into distinct sources, namely, therisk that a distinct state will occur. Such hypothetical financialproducts are useful since the returns from more complicated securities,including real-world financial products, can be modeled as a linearcombination of the returns of the hypothetical financial products. See,e.g., R. Merton, Continuous-Time Finance (1990), pp. 441 ff. Thus, suchhypothetical financial products are frequently used today to provide thefundamental building blocks for analyzing more complex financialproducts.

In the past fifteen years, the growth in derivatives trading has alsobeen enormous. According to the Federal Reserve, the annualized growthrate in foreign exchange and interest rate derivatives turnover alone isstill running at about 20%. Corporations, financial institutions,farmers, and even national governments and agencies are all active inthe derivatives markets, typically to better manage asset and liabilityportfolios, hedge financial market risk, and minimize costs of capitalfunding. Money managers also frequently use derivatives to hedge andundertake economic exposure where there are inherent risks, such asrisks of fluctuation in interest rates, foreign exchange rates,convertibility into other securities or outstanding purchase offers forcash or exchange offers for cash or securities.

Derivatives are traded on exchanges, such as the option and futurescontracts traded on the Chicago Board of Trade (CBOT), as well asoff-exchange or over-the-counter (OTC) between two or more derivativecounterparties. On the major exchanges which operate trading activity inderivatives, orders are typically either transmitted electronically orvia open outcry in pits to member brokers who then execute the orders.These member brokers then usually balance or hedge their own portfolioof derivatives to suit their own risk and return criteria. Hedging iscustomarily accomplished by trading in the derivatives' underlyingsecurities or contracts (e.g., a futures contract in the case of anoption on that future) or in similar derivatives (e.g., futures expiringin different calendar months). For OTC derivatives, brokers or dealerscustomarily seek to balance their active portfolios of derivatives inaccordance with the trader's risk management guidelines andprofitability criteria.

Broadly speaking then, there are two widely utilized means by whichderivatives are currently traded: (1) order-matching and (2) principalmarket making. Order matching is a model followed by exchanges such asthe CBOT or the Chicago Mercantile Exchange and some newer onlineexchanges. In order matching, the exchange coordinates the activities ofbuyers and sellers so that “bids” to buy (i.e., demand) can be pairedoff with “offers” to sell (i.e., supply). Orders may be matched bothelectronically and through the primary market making activities of theexchange members. Typically, the exchange itself takes no market riskand covers its own cost of operation by selling memberships to brokers.Member brokers may take principal positions, which are often hedgedacross their portfolios.

In principal market making, a bank or brokerage firm, for example,establishes a derivatives trading operation, capitalizes it, and makes amarket by maintaining a portfolio of derivatives and underlyingpositions. The market maker usually hedges the portfolio on a dynamicbasis by continually changing the composition of the portfolio as marketconditions change. In general, the market maker strives to cover itscost of operation by collecting a bid-offer spread and through the scaleeconomies obtained by simultaneously hedging a portfolio of positions.As the market maker takes significant market risk, its counterpartiesare exposed to the risk that it may go bankrupt. Additionally, while intheory the principal market making activity could be done over a widearea network, in practice derivatives trading is today usuallyaccomplished via the telephone. Often, trades are processed laboriously,with many manual steps required from the front office transaction to theback office processing and clearing.

In theory—that is, ignoring very real transaction costs (describedbelow)—derivatives trading is, in the language of game theory, a “zerosum” game. One counterparty's gain on a transaction should be exactlyoffset by the corresponding counterparty's loss, assuming there are notransaction costs. In fact, it is the zero sum nature of the derivativesmarket which first allowed the well-known Black-Scholes pricing model tobe formulated by noting that a derivative such as an option could bepaired with an exactly offsetting position in the underlying security soas to eliminate market risk over short periods of time. It is this “noarbitrage” feature, which allows market participants, usingsophisticated valuation models, to mitigate market risk by continuallyadjusting their portfolios. Stock markets, by contrast, do not have thiszero sum feature, as the total stock or value of the market fluctuatesdue to factors such as interest rates and expected corporate earnings,which are “external” to the market in the sense that they cannot readilybe hedged.

The return to a trader of a traditional derivative product is, in mostcases, largely determined by the value of the underlying security,asset, liability or claim on which the derivative is based. For example,the value of a call option on a stock, which gives the holder the rightto buy the stock at some future date at a fixed strike price, variesdirectly with the price of the underlying stock. In the case ofnon-financial derivatives such as reinsurance contracts, the value ofthe reinsurance contract is affected by the loss experience on theunderlying portfolio of insured claims. The prices of traditionalderivative products are usually determined by supply and demand for thederivative based on the value of the underlying security (which isitself usually determined by supply and demand, or, as in the case ofinsurance, by events insured by the insurance or reinsurance contract).

Currently, the costs of trading derivative securities (both on and offthe exchanges) and transferring insurance risk are considered to be highfor a number of reasons, including:

-   -   (1) Credit Risk: A counterparty to a derivatives (or insurance        contract) transaction typically assumes the risk that its        counterparty will go bankrupt during the life of the derivatives        (or insurance) contract. Margin requirements, credit monitoring,        and other contractual devices, which may be costly, are        customarily employed to manage derivatives and insurance        counterparty credit risk.    -   (2) Regulatory Requirements: Regulatory bodies, such as the        Federal Reserve, Comptroller of the Currency, the Commodities        Futures Trading Commission, and international bodies that        promulgate regulations affecting global money center banks        (e.g., Basle Committee guidelines) generally require        institutions dealing in derivatives to meet capital requirements        and maintain risk management systems. These requirements are        considered by many to increase the cost of capital and barriers        to entry for some entrants into the derivatives trading        business, and thus to increase the cost of derivatives        transactions for both dealers and end users. In the United        States, state insurance regulations also impose requirements on        the operations of insurers, especially in the property-casualty        lines where capital demands may be increased by the requirement        that insurers reserve for future losses without regard to        interest rate discount factors.    -   (3) Liquidity: Derivatives traders typically hedge their        exposures throughout the life of the derivatives contract.        Effective hedging usually requires that an active or liquid        market exist, throughout the life of the derivative contract,        for both the underlying security and the derivative. Frequently,        especially in periods of financial market shocks and        disequilibria, liquid markets do not exist to support a        well-functioning derivatives market.

(4) Transaction Costs: Dynamic hedging of derivatives often requirescontinual transactions in the market over the life of the derivative inorder to reduce, eliminate, and manage risk for a derivative orportfolio of derivative securities. This usually means paying bid-offersspreads for each hedging transaction, which can add significantly to theprice of the derivative security at inception compared to itstheoretical price in absence of the need to pay for such spreads andsimilar transaction costs.

-   -   (5) Settlement and Clearing Costs: The costs of executing,        electronically booking, clearing, and settling derivatives        transactions can be large, sometimes requiring analytical and        database software systems and personnel knowledgeable in such        transactions. While a goal of many in the securities processing        industry is to achieve “straight-through-processing” of        derivatives transactions, many derivatives counterparties        continue to manage the processing of these transactions using a        combination of electronic and manual steps which are not        particularly integrated or automated and therefore add to costs.

(6) Event Risk: Most traders understand effective hedging of derivativestransactions to require markets to be liquid and to exhibit continuouslyfluctuating prices without sudden and dramatic “gaps.” During periods offinancial crises and disequilibria, it is not uncommon to observedramatic repricing of underlying securities by 50% or more in a periodof hours. The event risk of such crises and disequilibria are thereforecustomarily factored into derivatives prices by dealers, which increasesthe cost of derivatives in excess of the theoretical prices indicated byderivatives valuation models. These costs are usually spread across allderivatives users.

(7) Model Risk: Derivatives contracts can be quite difficult to value,especially those involving interest rates or features which allow acounterparty to make decisions throughout the life of the derivative(e.g., American options allow a counterparty to realize the value of thederivative at any time during its life). Derivatives dealers willtypically add a premium to derivatives prices to insure against thepossibility that the valuation models may not adequately reflect marketfactors or other conditions throughout the life of the contract. Inaddition, risk management guidelines may require firms to maintainadditional capital supporting a derivatives dealing operation wheremodel risk is determined to be a significant factor. Model risk has alsobeen a large factor in well-known cases where complicated securitiesrisk management systems have provided incorrect or incompleteinformation, such as the Joe Jett/Kidder Peabody losses of 1994.

-   -   (8) Asymmetric Information: Derivatives dealers and market        makers customarily seek to protect themselves from        counterparties with superior information. Bid-offer spreads for        derivatives therefore usually reflect a built-in insurance        premium for the dealer for transactions with counterparties with        superior information, which can lead to unprofitable        transactions. Traditional insurance markets also incur costs due        to asymmetric information. In property-casualty lines, the        direct writer of the insurance almost always has superior        information regarding the book of risks than does the assuming        reinsurer. Much like the market maker in capital markets, the        reinsurer typically prices its informational disadvantage into        the reinsurance premiums.

(9) Incomplete Markets: Traditional capital and insurance markets areoften viewed as incomplete in the sense that the span of contingentclaims is limited, i.e., the markets may not provide opportunities tohedge all of the risks for which hedging opportunities are sought. As aconsequence, participants typically either bear risk inefficiently oruse less than optimal means to transfer or hedge against risk. Forexample, the demand by some investors to hedge inflation risk hasresulted in the issuance by some governments of inflation-linked bondswhich have coupons and principal amounts linked to Consumer Price Index(CPI) levels. This provides a degree of insurance against inflationrisk. However, holders of such bonds frequently make assumptions as tothe future relationship between real and nominal interest rates. Animperfect correlation between the contingent claim (in this case,inflation-linked bond) and the contingent event (inflation) gives riseto what traders call “basis risk,” which is risk that, in today'smarkets, cannot be perfectly insured or hedged.

Currently, transaction costs are also considerable in traditionalinsurance and reinsurance markets. In recent years, considerable efforthas been expended in attempting to securitize insurance risk such asproperty-casualty catastrophe risk. Traditional insurance andreinsurance markets in many respects resemble principal market-makersecurities markets and suffer from many of the same shortcomings andincur similar costs of operation. Typically, risk is physicallytransferred contractually, credit status of counterparties is monitored,and sophisticated risk management systems are deployed and maintained.Capitalization levels to support insurance portfolios of risky assetsand liabilities may be dramatically out of equilibrium at any given timedue to price stickiness, informational asymmetries and costs, andregulatory constraints. In short, the insurance and reinsurance marketstend to operate according to the same market mechanisms that haveprevailed for decades, despite large market shocks such as the Lloydscrisis in the late 1980's and early 1990's.

Accordingly, a driving force behind all the contributors to the costs ofderivatives and insurance contracts is the necessity or desirability ofrisk management through dynamic hedging or contingent claim replicationin continuous, liquid, and informationally fair markets. Hedging is usedby derivatives dealers to reduce their exposure to excessive market riskwhile making transaction fees to cover their cost of capital and ongoingoperations; and effective hedging requires liquidity.

Recent patents have addressed the problem of financial market liquidityin the context of an electronic order-matching systems (e.g., U.S. Pat.No. 5,845,266). The principal techniques disclosed to enhance liquidityare to increase participation and traded volume in the system and tosolicit trader preferences about combinations of price and quantity fora particular trade of a security. There are shortcomings to thesetechniques, however. First, these techniques implement order-matchingand limit order book algorithms, which can be and are effectivelyemployed in traditional “brick and mortar” exchanges. Their electronicimplementation, however, primarily serves to save on transportation andtelecommunication charges. No fundamental change is contemplated tomarket structure for which an electronic network may be essential.Second, the disclosed techniques appear to enhance liquidity at theexpense of placing large informational burdens on the traders (bysoliciting preferences, for example, over an entire price-quantitydemand curve) and by introducing uncertainty as to the exact price atwhich a trade has been transacted or is “filled.” Finally, theseelectronic order matching systems contemplate a traditional counterpartypairing, which means physical securities are frequently transferred,cleared, and settled after the counterparties are identified andmatched. In other words, techniques disclosed in the context ofelectronic order-matching systems are technical elaborations to thebasic problem of how to optimize the process of matching arrays of bidsand offers.

Patents relating to derivatives, such as U.S. Pat. No. 4,903,201,disclose an electronic adaptation of current open-outcry or ordermatching exchanges for the trading of futures is disclosed. Anotherrecent patent, U.S. Pat. No. 5,806,048, relates to the creation ofopen-end mutual fund derivative securities to provide enhanced liquidityand improved availability of information affecting pricing. This patent,however, does not contemplate an electronic derivatives exchange whichrequires the traditional hedging or replicating portfolio approach tosynthesizing the financial derivatives. Similarly, U.S. Pat. No.5,794,207 proposes an electronic means of matching buyers' bids andsellers' offers, without explaining the nature of the economic priceequilibria achieved through such a market process.

SUMMARY OF THE INVENTION

The present invention is directed to systems and methods of trading, andfinancial products, having a goal of reducing transaction costs formarket participants who hedge against or otherwise make investments incontingent claims relating to events of economic significance. Theclaims are contingent in that their payout or return depends on theoutcome of an observable event with more than one possible outcome. Theevents are of economic significance in that an investor or trader in acontingent claim typically is not economically indifferent to theoutcome of the event, even if the investor or trader has not invested inor traded a contingent claim relating to the event.

Intended users of preferred and other embodiments are typicallyinstitutional investors, such as financial institutions including banks,investment banks, primary insurers and reinsurers, and corporatetreasurers. Users can also include any individual or entity with a needfor risk allocation services. As used in this specification, the terms“user,” “trader” and “investor” are used interchangeably to mean anyinstitution, individual or entity that desires to trade or invest incontingent claims or other financial products described in thisspecification.

The contingent claims pertaining to an event have a trading period inorder to finalize a return for each defined state, which includes anoutcome or set of outcomes for the event, and another period forobserving the event upon which the contingent claim is based. Thereturns to the contingent claims of the present invention adjust duringthe trading period with changes in the distribution of amounts investedin each of the states. The returns payable for each of the states arefinalized after the conclusion of each relevant trading period. In apreferred embodiment, the total amount invested, less a transaction feeto an exchange, is equal to the total amount of the payouts. In otherwords, in theory, the returns on all of the contingent claimsestablished during a particular trading period and pertaining to aparticular event are essentially zero sum, as are the traditionalderivatives markets.

The process by which returns are finalized in the present invention isdemand-based, and does not in any substantial way depend on supply. Bycontrast, traditional markets set prices through the interaction ofsupply and demand by crossing bids to buy and offers to sell(“bid/offer”). The demand-based contingent claim mechanism of thepresent invention sets returns by financing returns to successfulinvestments with losses from unsuccessful investments. Thus, in apreferred embodiment, the returns to successful investments aredetermined by the total and relative amounts of all investments placedon each of the defined states for the specified observable event.

As used in this specification, the term “contingent claim” shall havethe meaning customarily ascribed to it in the securities, trading,insurance and economics communities. “Contingent claims” thus includes,for example, stocks, bonds and other such securities, derivativesecurities, insurance contracts and reinsurance agreements, and anyother financial products, instruments, contracts, assets, or liabilitieswhose value depends upon or reflects economic risk due to the occurrenceof future, real-world events. These events may be financial-relatedevents, such as changes in interest rates, or non-financial-relatedevents such as changes in weather conditions, demand for electricity,and fluctuations in real estate prices. Contingent claims also includeall economic or financial interests, whether already traded or not yettraded, which have or reflect inherent risk or uncertainty due to theoccurrence of future real-world events. Examples of contingent claims ofeconomic or financial interest which are not yet traded on traditionalmarkets are financial products having values that vary with thefluctuations in corporate earnings or changes in real estate values andrentals. The term “contingent claim” as used in this specificationencompasses both hypothetical financial products of the Arrow-Debreuvariety, as well as any risky asset, contract or product which can beexpressed as a combination or portfolio of the hypothetical financialproducts.

For the purposes of this specification, an “investment” in or “trade” ofa contingent claim is the act of putting an amount (in the units ofvalue defined by the contingent claim) at risk, with a financial returndepending on the outcome of an event of economic significance underlyingthe group of contingent claims pertaining to that event.

“Derivative security” (used interchangeably with “derivative”) also hasa meaning customarily ascribed to it in the securities, trading,insurance and economics communities. This includes a security orcontract whose value depends on such factors as the value of anunderlying security, index, asset or liability, or on a feature of suchan underlying security, such as interest rates or convertibility intosome other security. A derivative security is one example of acontingent claim as defined above. Financial futures on stock indicessuch as the S&P 500 or options to buy and sell such futures contractsare highly popular exchange-traded financial derivatives. Aninterest-rate swap, which is an example of an off-exchange derivative,is an agreement between two counterparties to exchange series ofcashflows based on underlying factors, such as the London InterbankOffered Rate (LIBOR) quoted daily in London for a large number offoreign currencies. Like the exchange-traded futures and options,off-exchange agreements can fluctuate in value with the underlyingfactors to which they are linked or derived. Derivatives may also betraded on commodities, insurance events, and other events, such as theweather.

In this specification, the function for computing and allocating returnsto contingent claims is termed the Demand Reallocation Function (DRF). ADRF is demand-based and involves reallocating returns to investments ineach state after the outcome of the observable event is known in orderto compensate successful investments from losses on unsuccessfulinvestments (after any transaction or exchange fee). Since an adjustablereturn based on variations in amounts invested is a key aspect of theinvention, contingent claims implemented using a DRF will be referred toas demand-based adjustable return (DBAR) contingent claims.

Preferred features of a trading system for a group of DBAR contingentclaims (i.e., group of claims pertaining to the same event) include thefollowing: (1) an entire distribution of states is open for investment,not just a single price as in the traditional markets; (2) returns areadjustable and determined mathematically based on invested amounts ineach of the states available for investment, (3) invested amounts arepreferably non-decreasing (as explained below), providing a commitmentof offered liquidity to the market over the distribution of states, and(4) information is available in real-time across the distribution ofstates, including, in particular, information on the amounts investedacross the distribution of all states (commonly known as a “limit orderbook”). Other consequences of preferred embodiments of the presentinvention include (1) elimination of order-matching or crossing of thebid and offer sides of the market; (2) reduction of the need for amarket maker to conduct dynamic hedging and risk management; and (3)more opportunities for hedging and insuring events of economicsignificance (i.e., greater market “completeness”).

Other preferred embodiments of the present invention can accommodaterealization of profits and losses by traders at multiple points beforeall of the criteria for terminating a group of contingent claims areknown. This is accomplished by arranging a plurality of trading periods,each having its own set of finalized returns. Profit or loss can berealized or “locked-in” at the end of each trading period, as opposed towaiting for the final outcome of the event on which the relevantcontingent claims are based. Such lock-in can be achieved by placinghedging investments in successive trading periods as the returns change,or adjust, from period to period. In this way, profit and loss can berealized on an evolving basis (limited only by the frequency and lengthof the periods), enabling traders to achieve the same or perhaps higherfrequency of trading and hedging than available in traditional markets.

If desired, an issuer such as a corporation, investment bank,underwriter or other financial intermediary can create a security havingreturns that are driven in a comparable manner to the DBAR contingentclaims of the present invention. For example, a corporation may issue abond with returns that are linked to insurance risk. The issuer cansolicit trading and calculate the returns based on the amounts investedin contingent claims corresponding to each level or state of insurancerisks.

In a preferred embodiment of the present invention, changes in thereturn for investments in one state will affect the return oninvestments in another state in the same distribution of states for agroup of contingent claims. Thus, traders' returns will depend not onlyon the actual outcome of a real-world, observable event but also ontrading choices from among the distribution of states made by othertraders. This aspect of DBAR markets, in which returns for one state areaffected by changes in investments in another state in the samedistribution, allows for the elimination of order-crossing and dynamicmarket maker hedging. Price-discovery in preferred embodiments of thepresent invention can be supported by a one-way market (i.e., demand,not supply) for DBAR contingent claims. By structuring derivatives andinsurance trading according to DBAR principles, the high costs oftraditional order matching and principal market making market structurescan be reduced substantially. Additionally, a market implemented bysystems and methods of the present invention is especially amenable toelectronic operation over a wide network, such as the Internet.

In its preferred embodiments, the present invention mitigatesderivatives transaction costs found in traditional markets due todynamic hedging and order matching. A preferred embodiment of thepresent invention provides a system for trading contingent claimsstructured under DBAR principles, in which amounts invested in on eachstate in a group of DBAR contingent claims are reallocated fromunsuccessful investments, under defined rules, to successful investmentsafter the deduction of exchange transaction fees. In particular, theoperator of such a system or exchange provides the physical plant andelectronic infrastructure for trading to be conducted, collects andaggregates investments, calculates the returns that result from suchinvestments, and then allocates to the successful investments returnsthat are financed by the unsuccessful investments, after deducting atransaction fee for the operation of the system.

In preferred embodiments, where the successful investments are financedwith the losses from unsuccessful investments, returns on all trades arecorrelated and traders make investments against each other as well asassuming the risk of chance outcomes. All traders for a group of DBARcontingent claims depending on a given event become counterparties toeach other, leading to a mutualization of financial interests.Furthermore, in preferred embodiments of the present invention,projected returns prevailing at the time an investment is made may notbe the same as the final payouts or returns after the outcome of therelevant event is known.

Traditional derivatives markets by contrast, operate largely under ahouse “banking” system. In this system, the market-maker, whichtypically has the function of matching buyers and sellers, customarilyquotes a price at which an investor may buy or sell. If a given investorbuys or sells at the price, the investor's ultimate return is based uponthis price, i.e., the price at which the investor later sells or buysthe original position, along with the original price at which theposition was traded, will determine the investor's return. As themarket-maker may not be able perfectly to offset buy and sell orders atall times or may desire to maintain a degree of risk in the expectationof returns, it will frequently be subject to varying degrees of marketrisk (as well as credit risk, in some cases). In a traditionalderivatives market, market-makers which match buy and sell orderstypically rely upon actuarial advantage, bid-offer spreads, a largecapital base, and “coppering” or hedging (risk management) to minimizethe chance of bankruptcy due to such market risk exposures.

Each trader in a house banking system typically has only a singlecounterparty—the market-maker, exchange, or trading counterparty (in thecase, for example, of over-the-counter derivatives). By contrast,because a market in DBAR contingent claims may operate according toprinciples whereby unsuccessful investments finance the returns onsuccessful investments, the exchange itself is exposed to reduced riskof loss and therefore has reduced need to transact in the market tohedge itself. In preferred embodiments of DBAR contingent claims of thepresent invention, dynamic hedging or bid-offer crossing by the exchangeis generally not required, and the probability of the exchange ormarket-maker going bankrupt may be reduced essentially to zero. Such asystem distributes the risk of bankruptcy away from the exchange ormarket-maker and among all the traders in the system. The system as awhole provides a great degree of self-hedging and substantial reductionof the risk of market failure for reasons related to market risk. A DBARcontingent claim exchange may also “self-clearing” and require littleclearing infrastructure (such as clearing agents, custodians,nostro/vostro bank accounts, and transfer and register agents). Aderivatives trading system or exchange structured according to DBARcontingent claim principles therefore offers many advantages overcurrent derivatives markets governed by house banking principles.

The present invention also differs from electronic or parimutuel bettingsystems disclosed in the prior art (e.g., U.S. Pat. Nos. 5,873,782 and5,749,785). In betting systems or games of chance, in the absence of awager the bettor is economically indifferent to the outcome (assumingthe bettor does not own the casino or the racetrack or breed the racinghorses, for example). The difference between games of chance and eventsof economic significance is well known and understood in financialmarkets.

In summary, the present invention provides systems and methods forconducting demand-based trading. A preferred embodiment of a method ofthe present invention for conducting demand-based trading includes thesteps of (a) establishing a plurality of defined states and a pluralityof predetermined termination criteria, wherein each of the definedstates corresponds to at least one possible outcome of an event ofeconomic significance; (b) accepting investments of value units by aplurality of traders in the defined states; and (c) allocating a payoutto each investment. The allocating step is responsive to the totalnumber of value units invested in the defined states, the relativenumber of value units invested in each of the defined states, and theidentification of the defined state that occurred upon fulfillment ofall of the termination criteria.

An additional preferred embodiment of a method for conductingdemand-based trading also includes establishing, accepting, andallocating steps. The establishing step in this embodiment includesestablishing a plurality of defined states and a plurality ofpredetermined termination criteria. Each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled. The accepting stepincludes accepting investments of value units by multiple traders in thedefined states. The allocating step includes allocating a payout to eachinvestment. This allocating step is responsive to the total number ofvalue units invested in the defined states, the relative number of valueunits invested in each of the defined states, and the identification ofthe defined state that occurred upon fulfillment of all of thetermination criteria.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the payout to each investment in each of thedefined states that did not occur upon fulfillment of all of thetermination criteria is zero, and the sum of the payouts to all of theinvestments is not greater than the value of the total number of thevalue units invested in the defined states. In a further preferredembodiment, the sum of the values of the payouts to all of theinvestments is equal to the value of all of the value units invested indefined states, less a fee.

In preferred embodiments of a method for conducting demand-basedtrading, at least one investment of value units designates a set ofdefined states and a desired return-on-investment from the designatedset of defined states. In these preferred embodiments, the allocatingstep is further responsive to the desired return-on-investment from thedesignated set of defined states.

In another preferred embodiment of a method for conducting demand-basedtrading, the method further includes the step of calculatingCapital-At-Risk for at least one investment of value units by at leastone trader. In alternative further preferred embodiments, the step ofcalculating Capital-At-Risk includes the use of the Capital-At-RiskValue-At-Risk method, the Capital-At-Risk Monte Carlo Simulation method,or the Capital-At-Risk Historical Simulation method.

In preferred embodiments of a method for conducting demand-basedtrading, the method further includes the step of calculatingCredit-Capital-At-Risk for at least one investment of value units by atleast one trader. In alternative further preferred embodiments, the stepof calculating Credit-Capital-At-Risk includes the use of theCredit-Capital-At-Risk Value-At-Risk method, the Credit-Capital-At-RiskMonte Carlo Simulation method, or the Credit-Capital-At-Risk HistoricalSimulation method.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, at least one investment of value units is amulti-state investment that designates a set of defined states. In afurther preferred embodiment, at least one multi-state investmentdesignates a set of desired returns that is responsive to the designatedset of defined states, and the allocating step is further responsive tothe set of desired returns. In a further preferred embodiment, eachdesired return of the set of desired returns is responsive to a subsetof the designated set of defined states. In an alternative preferredembodiment, the set of desired returns approximately corresponds toexpected returns from a set of defined states of a prespecifiedinvestment vehicle such as, for example, a particular call option.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the allocating step includes the steps of (a)calculating the required number of value units of the multi-stateinvestment that designates a set of desired returns, and (b)distributing the value units of the multi-state investment thatdesignates a set of desired returns to the plurality of defined states.In a further preferred embodiment, the allocating step includes the stepof solving a set of simultaneous equations that relate traded amounts tounit payouts and payout distributions; and the calculating step and thedistributing step are responsive to the solving step.

In preferred embodiments of a method for conducting demand-based tradingof the present invention, the solving step includes the step of fixedpoint iteration. In further preferred embodiments, the step of fixedpoint iteration includes the steps of (a) selecting an equation of theset of simultaneous equations described above, the equation having anindependent variable and at least one dependent variable; (b) assigningarbitrary values to each of the dependent variables in the selectedequation; (c) calculating the value of the independent variable in theselected equation responsive to the currently assigned values of eachthe dependent variables; (d) assigning the calculated value of theindependent variable to the independent variable; (e) designating anequation of the set of simultaneous equations as the selected equation;and (f) sequentially performing the calculating the value step, theassigning the calculated value step, and the designating an equationstep until the value of each of the variables converges.

A preferred embodiment of a method for estimating state probabilities ina demand-based trading method of the present invention includes thesteps of: (a) performing a demand-based trading method having aplurality of defined states and a plurality of predetermined terminationcriteria, wherein an investment of value units by each of a plurality oftraders is accepted in at least one of the defined states, and at leastone of these defined states corresponds to at least one possible outcomeof an event of economic significance; (b) monitoring the relative numberof value units invested in each of the defined states; and (c)estimating, responsive to the monitoring step, the probability that aselected defined state will be the defined state that occurs uponfulfillment of all of the termination criteria.

An additional preferred embodiment of a method for estimating stateprobabilities in a demand-based trading method also includes performing,monitoring, and estimating steps. The performing step includesperforming a demand-based trading method having a plurality of definedstates and a plurality of predetermined termination criteria, wherein aninvestment of value units by each of a plurality of traders is acceptedin at least one of the defined states; and wherein each of the definedstates corresponds to a possible state of a selected financial productwhen each of the termination criteria is fulfilled. The monitoring stepincludes monitoring the relative number of value units invested in eachof the defined states. The estimating step includes estimating,responsive to the monitoring step, the probability that a selecteddefined state will be the defined state that occurs upon fulfillment ofall of the termination criteria.

A preferred embodiment of a method for promoting liquidity in ademand-based trading method of the present invention includes the stepof performing a demand-based trading method having a plurality ofdefined states and a plurality of predetermined termination criteria,wherein an investment of value units by each of a plurality of tradersis accepted in at least one of the defined states and wherein anyinvestment of value units cannot be withdrawn after acceptance. Each ofthe defined states corresponds to at least one possible outcome of anevent of economic significance. A further preferred embodiment of amethod for promoting liquidity in a demand-based trading method includesthe step of hedging. The hedging step includes the hedging of a trader'sprevious investment of value units by making a new investment of valueunits in one or more of the defined states not invested in by theprevious investment.

An additional preferred embodiment of a method for promoting liquidityin a demand-based trading method includes the step of performing ademand-based trading method having a plurality of defined states and aplurality of predetermined termination criteria, wherein an investmentof value units by each of a plurality of traders is accepted in at leastone of the defined states and wherein any investment of value unitscannot be withdrawn after acceptance, and each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled. A further preferredembodiment of such a method for promoting liquidity in a demand-basedtrading method includes the step of hedging. The hedging step includesthe hedging of a trader's previous investment of value units by making anew investment of value units in one or more of the defined states notinvested in by the previous investment.

A preferred embodiment of a method for conducting quasi-continuousdemand-based trading includes the steps of: (a) establishing a pluralityof defined states and a plurality of predetermined termination criteria,wherein each of the defined states corresponds to at least one possibleoutcome of an event; (b) conducting a plurality of trading cycles,wherein each trading cycle includes the step of accepting, during apredefined trading period and prior to the fulfillment of all of thetermination criteria, an investment of value units by each of aplurality of traders in at least one of the defined states; and (c)allocating a payout to each investment. The allocating step isresponsive to the total number of the value units invested in thedefined states during each of the trading periods, the relative numberof the value units invested in each of the defined states during each ofthe trading periods, and an identification of the defined state thatoccurred upon fulfillment of all of the termination criteria In afurther preferred embodiment of a method for conducting quasi-continuousdemand-based trading, the predefined trading periods are sequential anddo not overlap.

Preferred embodiments of the system of the present invention involve theuse of electronic technologies, such as computers, computerizeddatabases and telecommunications systems, to implement methods forconducting demand-based trading of the present invention.

A preferred embodiment of a system of the present invention forconducting demand-based trading includes (a) means for accepting, priorto the fulfillment of all predetermined termination criteria,investments of value units by a plurality of traders in at least one ofa plurality of defined states, wherein each of the defined statescorresponds to at least one possible outcome of an event of economicsignificance; and (b) means for allocating a payout to each investment.This allocation is responsive to the total number of value unitsinvested in the defined states, the relative number of value unitsinvested in each of the defined states, and the identification of thedefined state that occurred upon fulfillment of all of the terminationcriteria.

An additional preferred embodiment of a system of the present inventionfor conducting demand-based trading includes (a) means for accepting,prior to the fulfillment of all predetermined termination criteria,investments of value units by a plurality of traders in at least one ofa plurality of defined states, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled; and (b) means forallocating a payout to each investment. This allocation is responsive tothe total number of value units invested in the defined states, therelative number of value units invested in each of the defined states,and the identification of the defined state that occurred uponfulfillment of all of the termination criteria.

A preferred embodiment of a demand-based trading apparatus of thepresent invention includes (a) an interface processor communicating witha plurality of traders and a market data system; and (b) a demand-basedtransaction processor, communicating with the interface processor andhaving a trade status database. The demand-based transaction processormaintains, responsive to the market data system and to a demand-basedtransaction with one of the plurality of traders, the trade statusdatabase, and processes, responsive to the trade status database, thedemand-based transaction.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includes(a) establishing a contingent claim having a plurality of definedstates, a plurality of predetermined termination criteria, and at leastone trading period, wherein each of the defined states corresponds to atleast one possible outcome of an event of economic significance; (b)recording, responsive to the demand-based transaction, an investment ofvalue units by one of the plurality of traders in at least one of theplurality of defined states; (c) calculating, responsive to the totalnumber of the value units invested in the plurality of defined statesduring each trading period and responsive to the relative number of thevalue units invested in each of the plurality of defined states duringeach trading period, finalized returns at the end of each tradingperiod; and (d) determining, responsive to an identification of thedefined state that occurred upon the fulfillment of all of thetermination criteria and to the finalized returns, payouts to each ofthe plurality of traders; and processing the demand-based transactionincludes accepting, during the trading period, the investment of valueunits by one of the plurality of traders in at least one of theplurality of defined states;

In an alternative further preferred embodiment of a demand-based tradingapparatus of the present invention, maintaining the trade statusdatabase includes (a) establishing a contingent claim having a pluralityof defined states, a plurality of predetermined termination criteria,and at least one trading period, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the termination criteria is fulfilled; (b) recording, responsiveto the demand-based transaction, an investment of value units by one ofthe plurality of traders in at least one of the plurality of definedstates; (c) calculating, responsive to the total number of the valueunits invested in the plurality of defined states during each tradingperiod and responsive to the relative number of the value units investedin each of the plurality of defined states during each trading period,finalized returns at the end of each trading period; and (d)determining, responsive to an identification of the defined state thatoccurred upon the fulfillment of all of the termination criteria and tothe finalized returns, payouts to each of the plurality of traders; andprocessing the demand-based transaction includes accepting, during thetrading period, the investment of value units by one of the plurality oftraders in at least one of the plurality of defined states;

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includescalculating return estimates; and processing the demand-basedtransaction includes providing, responsive to the demand-basedtransaction, the return estimates.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, maintaining the trade status database includescalculating risk estimates; and processing the demand-based transactionincludes providing, responsive to the demand-based transaction, the riskestimates.

In further preferred embodiments of a demand-based trading apparatus ofthe present invention, the demand-based transaction includes amulti-state investment that specifies a desired payout distribution anda set of constituent states; and maintaining the trade status databaseincludes allocating, responsive to the multi-state investment, valueunits to the set of constituent states to create the desired payoutdistribution.

An object of the present invention is to provide systems and methods tosupport and facilitate a market structure for contingent claims relatedto observable events of economic significance, which includes one ormore of the following advantages, in addition to those described above:

-   -   1. ready implementation and support using electronic computing        and networking technologies;    -   2. reduction or elimination of the need to match bids to buy        with offers to sell in order to create a market for derivatives;    -   3. reduction or elimination of the need for a derivatives        intermediary to match bids and offers;    -   4. mathematical and consistent calculation of returns based on        demand for contingent claims;    -   5. increased liquidity;    -   6. statistical diversification of credit risk through the        mutualization of multiple derivatives counterparties;    -   7. improved scalability by reducing the traditional linkage        between the method of pricing for contingent claims and the        quantity of the underlying claims available for investment;    -   8. increased price transparency;    -   9. improved efficiency of information aggregation mechanisms;    -   10. reduction of event risk, such as the risk of discontinuous        market events such as crashes;    -   11. opportunities for binding offers of liquidity to the market;        and    -   12. reduced incentives for strategic behavior by traders.

A further object of the present invention is to provide systems andmethods for the electronic exchange of contingent claims related toobservable events of economic significance, which includes one or moreof the following advantages:

-   -   1. reduced transaction costs, including settlement and clearing        costs, associated with derivatives transactions and insurable        claims;    -   2. reduced dependence on complicated valuation models for        trading and risk management of derivatives;    -   3. reduced need for an exchange or market maker to manage market        risk by hedging;    -   4. increased availability to traders of accurate and up-to-date        information on the trading of contingent claims, including        information regarding the aggregate amounts invested across all        states of events of economic significance, and including over        varying time periods;    -   5. reduced exposure of the exchange to credit risk;    -   6. increased availability of information on credit risk and        market risk borne by traders of contingent claims;    -   7. increased availability of information on marginal returns        from trades and investments that can be displayed        instantaneously after the returns adjust during a trading        period;    -   8. reduced need for a derivatives intermediary or exchange to        match bids and offers; and    -   9. increased ability to customize demand-based adjustable return        (DBAR) payouts to permit replication of traditional financial        products and their derivatives.

Additional objects and advantages of the invention are set forth in partin the description which follows, and in part are obvious from thedescription, or may be learned by practice of the invention. The objectsand advantages of the invention may also be realized and attained bymeans of the instrumentalities, systems, methods and steps set forth inthe appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

The accompanying drawings, which are incorporated in and from a part ofthe specification, illustrate the embodiments of the present inventionand, together with the description, serve to explain the principles ofthe invention.

FIG. 1 is a schematic view of various forms of telecommunicationsbetween DBAR trader clients and a preferred embodiment of a DBARcontingent claims exchange implementing the present invention.

FIG. 2 is a schematic view of a central controller of a preferredembodiment of a DBAR contingent claims exchange network architectureimplementing the present invention.

FIG. 3 is a schematic depiction of the trading process on a preferredembodiment of a DBAR contingent claims exchange.

FIG. 4 depicts data storage devices of a preferred embodiment of a DBARcontingent claims exchange.

FIG. 5 is a flow diagram illustrating the processes of a preferredembodiment of DBAR contingent claims exchange in executing a DBAR rangederivatives investment.

FIG. 6 is an illustrative HTML interface page of a preferred embodimentof a DBAR contingent claims exchange.

FIG. 7 is a schematic view of market data flow to a preferred embodimentof a DBAR contingent claims exchange.

FIG. 8 is an illustrative graph of the implied liquidity effects for agroup of DBAR contingent claims.

FIG. 9 a is a schematic representation of a traditional interest rateswap transaction.

FIG. 9 b is a schematic of investor relationships for an illustrativegroup of DBAR contingent claims.

FIG. 9 c shows a tabulation of credit ratings and margin trades for eachinvestor in to an illustrative group of DBAR contingent claims.

FIG. 10 is a schematic view of a feedback process for a preferredembodiment of DBAR contingent claims exchange.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

This Detailed Description of Preferred Embodiments is organized intoseven sections. The first section provides an overview of systems andmethods for trading or investing in groups of DBAR contingent claims.The second section describes in detail some of the important features ofsystems and methods for trading or investing in groups of DBARcontingent claims. The third section of this Detailed Description ofPreferred Embodiments provides detailed descriptions of two preferredembodiments of the present invention: investments in a group of DBARcontingent claims, and investments in a portfolio of groups of suchclaims. The fourth section discusses methods for calculating risksattendant on investments in groups and portfolios of groups of DBARcontingent claims. The fifth section of this Detailed Descriptionaddresses liquidity and price/quantity relationships in preferredembodiments of systems and methods of the present invention. The sixthsection presents a detailed description of the figures accompanying thisspecification. The final section of this Detailed Description discussessome of the salient advantages of the methods and systems of the presentinvention.

More specifically, this Detailed Description of the PreferredEmbodiments is organized as follows:

1 Overview: Exchanges and Markets for DBAR Contingent claims

-   -   1.1 Exchange Design    -   1.2 Market Operation    -   1.3 Network Implementation

2 Features of DBAR Contingent claims

-   -   2.1 DBAR Contingent Claim Notation    -   2.2 Units of Investment and Payouts    -   2.3 Canonical Demand Reallocation Functions    -   2.4 Computing Investment Amounts to Achieve Desired Payouts    -   2.5 A Canonical DRF Example    -   2.6 Interest Considerations    -   2.7 Returns and Probabilities    -   2.8 Computations When Invested Amounts are Large

3 Examples of Groups of DBAR Contingent claims

-   -   3.1 DBAR Range Derivatives (including 21 examples)    -   3.2 DBAR Portfolios

4 Risk Calculations in Groups of DBAR Contingent claims

-   -   4.1 Market Risk        -   4.1.1 Capital-At-Risk Determinations        -   4.1.2 Capital-At-Risk Determinations Using Monte Carlo            Simulation Techniques        -   4.1.3 Capital-At-Risk Determinations Using Historical            Simulation Techniques    -   4.2 Credit Risk        -   4.2.1 Credit-Capital-At-Risk Determinations        -   4.2.2 Credit-Capital-At-Risk Determinations using Monte            Carlo Simulation Techniques        -   4.2.3 Credit-Capital-At-Risk Historical Simulation            Techniques

5 Liquidity and Price/Quantity Relationships

6 Detailed Description of the Drawings

7 Advantages of Preferred Embodiments

8 Technical Appendix

In this specification, including the description of preferredembodiments of the present invention, specific terminology will be usedfor the sake of clarity. However, the invention is not intended to belimited to the specific terms so used, and it is to be understood thateach specific term includes all equivalents.

1 OVERVIEW: EXCHANGES AND MARKETS FOR DBAR CONTINGENT CLAIMS 1.1Exchange Design

This section describes preferred methods for structuring DBAR contingentclaims and for designing an exchange for the trading of such claims. Thedesign of the exchange is important for effective contingent claimsinvestment in accordance with the present invention. Preferredembodiments of such systems include processes for establishing definedstates and allocating returns, as described below.

-   -   (a) Establishing Defined States: In a preferred embodiment, a        distribution of possible outcomes for an observable event is        partitioned into defined ranges or states. In a preferred        embodiment, one state always occurs because the states are        mutually exclusive and collectively exhaustive. Traders in such        an embodiment invest on their expectation of a return resulting        from the occurrence of a particular outcome within a selected        state. Such investments allow traders to hedge the possible        outcomes of real-world events of economic significance        represented by the states. In a preferred embodiment of a group        of DBAR contingent claims, unsuccessful trades or investments        finance the successful trades or investments. In such an        embodiment the states for a given contingent claim preferably        are defined in such a way that the states are mutually exclusive        and form the basis of a probability distribution, namely, the        sum of the probabilities of all the uncertain outcomes is unity.        For example, states corresponding to stock price closing values        can be established to support a group DBAR contingent claims by        partitioning the distribution of possible closing values for the        stock on a given future date into ranges. The distribution of        future stock prices, discretized in this way into defined        states, forms a probability distribution in the sense that each        state is mutually exclusive, and the sum of the probabilities of        the stock closing within each defined state at the given date is        unity.        -   In a preferred embodiment, traders can simultaneously invest            in selected multiple states within a given distribution,            without immediately breaking up their investment to fit into            each defined states selected for investment. Traders thus            may place multi-state investments in order to replicate a            desired distribution of returns from a group of contingent            claims. This may be accomplished in a preferred embodiment            of a DBAR exchange through the use of suspense accounts in            which multi-state investments are tracked and reallocated            periodically as returns adjust in response to amounts            invested during a trading period. At the end of a given            trading period, a multi-state investment may be reallocated            to achieve the desired distribution of payouts based upon            the final invested amounts across the distribution of            states. Thus, in such a preferred embodiment, the invested            amount allocated to each of the selected states, and the            corresponding respective returns, are finalized only at the            closing of the trading period. An example of a multi-state            investment illustrating the use of such a suspense account            is provided in Example 3.1.2, below.    -   (b) Allocating Returns: In a preferred embodiment of a group of        DBAR contingent claims according to the present invention,        returns for each state are specified. In such an embodiment,        while the amount invested for a given trade may be fixed, the        return is adjustable. Determination of the returns for a        particular state can be a simple function of the amount invested        in that state and the total amount invested for all of the        defined states for a group of contingent claims. However,        alternate preferred embodiments can also accommodate methods of        return determination that include other factors in addition to        the invested amounts. For example, in a group of DBAR contingent        claims where unsuccessful investments fund returns to successful        investments, the returns can be allocated based on the relative        amounts invested in each state and also on properties of the        outcome, such as the magnitude of the price changes in        underlying securities. An example in section 3.2 below        illustrates such an embodiment in the context of a securities        portfolio.

1.2 Market Operation

-   -   (a) Termination Criteria: In a preferred embodiment of a method        of the present invention, returns to investments in the        plurality of defined states are allocated after the fulfillment        of predetermined termination criteria. In preferred embodiments,        these criteria include the expiration of a “trading period” and        the determination of the outcome of the relevant event after an        “observation period.” In the trading period, traders invest on        their expectation of a return resulting from the occurrence of a        particular outcome within a selected defined state, such as the        state that IBM stock will close between 120 and 125 on Jul.        6, 1999. In a preferred embodiment, the duration of the trading        period is known to all participants; returns associated with        each state vary during the trading period with changes in        invested amounts; and returns are allocated based on the total        amount invested in all states relative to the amounts invested        in each of the states as at the end of the trading period.

The observation period can be provided as a time period during which thecontingent events are observed and the relevant outcomes determined forthe purpose of allocating returns. In a preferred embodiment, no tradingoccurs during the observation period.

The expiration date, or “expiration”, of a group of DBAR contingentclaims as used in this specification occurs when the terminationcriteria are fulfilled for that group of DBAR contingent claims. In apreferred embodiment, the expiration is the date, on or after theoccurrence of the relevant event, when the outcome is ascertained. Thisexpiration is similar to well-known expiration features in traditionaloptions or futures in which a future date, i.e., the expiration date, isspecified as the date upon which the value of the option or future willbe determined by reference to the value of the underlying financialproduct on the expiration date.

-   -   -   The duration of a contingent claim as defined for purposes            of this specification is simply the amount of time remaining            until expiration from any given reference date. A trading            start date (“TSD”) and a trading end date (“TED”), as used            in the specification, refer to the beginning and end of a            time period (“trading period”) during which traders can make            investments in a group of DBAR contingent claims. Thus, the            time during which a group of DBAR contingent claims is open            for investment or trading , i.e., the difference between the            TSD and TED, may be referred to as the trading period. In            preferred embodiments, there can be one or many trading            periods for a given expiration date, opening successively            through time. For example, one trading period's TED may            coincide exactly with the subsequent trading period's TSD,            or in other examples, trading periods may overlap.

The relationship between the duration of a contingent claim, the numberof trading periods employed for a given event, and the length and timingof the trading periods, can be arranged in a variety of ways to maximizetrading or achieve other goals. In preferred embodiments at least onetrading period occurs—that is, starts and ends—prior in time to theidentification of the outcome of the relevant event. In other words, inpreferred embodiments, the trading period will most likely temporarilyprecede the event defining the claim. This need not always be so, sincethe outcome of an event may not be known for some time thereby enablingtrading periods to end (or even start) subsequent to the occurrence ofthe event, but before its outcome is known.

A nearly continuous or “quasi-continuous” market can be made availableby creating multiple trading periods for the same event, each having itsown closing returns. Traders can make investments during successivetrading periods as the returns change. In this way, profits-and-lossescan be realized at least as frequently as in current derivativesmarkets. This is how derivatives traders currently are able to hedgeoptions, futures, and other derivatives trades. In preferred embodimentsof the present invention, traders may be able to realize profits and atvarying frequencies, including more frequently than daily.

-   -   (b) Market Efficiency and Fairness: Market prices reflect, among        other things, the distribution of information available to        segments of the participants transacting in the market. In most        markets, some participants will be better informed than others.        In house-banking or traditional markets, market makers protect        themselves from more informed counterparties by increasing their        bid-offer spreads.

In preferred embodiments of DBAR contingent claim markets, there may beno market makers as such who need to protect themselves. It maynevertheless be necessary to put in place methods of operation in suchmarkets in order to prevent manipulation of the outcomes underlyinggroups of DBAR contingent claims or the returns payable for variousoutcomes. One such mechanism is to introduce an element of randomness asto the time at which a trading period closes. Another mechanism tominimize the likelihood and effects of market manipulation is tointroduce an element of randomness to the duration of the observationperiod. For example, a DBAR contingent claim might settle against anaverage of market closing prices during a time interval that ispartially randomly determined, as opposed to a market closing price on aspecific day.

Additionally, in preferred embodiments incentives can be employed inorder to induce traders to invest earlier in a trading period ratherthan later. For example, a DRF may be used which allocates slightlyhigher returns to earlier investments in a successful state than laterinvestments in that state. Earlier investments may be valuable inpreferred embodiments since they work to enhance liquidity and promotemore uniformly meaningful price information during the trading period.

-   -   (c) Credit Risk: In preferred embodiments of a DBAR contingent        claims market, the dealer or exchange is substantially protected        from primary market risk by the fundamental principle underlying        the operation of the system—that returns to successful        investments are funded by losses from unsuccessful investments.        The credit risk in such preferred embodiments is distributed        among all the market participants. If, for example, leveraged        investments are permitted within a group of DBAR contingent        claims, it may not be possible to collect the leveraged        unsuccessful investments in order to distribute these amounts        among the successful investments.

In almost all such cases there exists, for any given trader within agroup of DBAR contingent claims, a non-zero possibility of default, orcredit risk. Such credit risk is, of course, ubiquitous to all financialtransactions facilitated with credit.

One way to address this risk is to not allow leveraged investmentswithin the group of DBAR contingent claims, which is a preferredembodiment of the system and methods of the present invention. In otherpreferred embodiments, traders in a DBAR exchange may be allowed to uselimited leverage, subject to real-time margin monitoring, includingcalculation of a trader's impact on the overall level of credit risk inthe DBAR system and the particular group of contingent claims. Theserisk management calculations should be significantly more tractable andtransparent than the types of analyses credit risk managers typicallyperform in conventional derivatives markets in order to monitorcounterparty credit risk.

An important feature of preferred embodiments of the present inventionis the ability to provide diversification of credit risk among all thetraders who invest in a group of DBAR contingent claims. In suchembodiments, traders make investments (in the units of value as definedfor the group) in a common distribution of states in the expectation ofreceiving a return if a given state is determined to have occurred. Inpreferred embodiments, all traders, through their investments in definedstates for a group of contingent claims, place these invested amountswith a central exchange or intermediary which, for each trading period,pays the returns to successful investments from the losses onunsuccessful investments. In such embodiments, a given trader has allthe other traders in the exchange as counterparties, effecting amutualization of counterparties and counterparty credit risk exposure.Each trader therefore assumes credit risk to a portfolio ofcounterparties rather than to a single counterparty.

Preferred embodiments of the DBAR contingent claim and exchange of thepresent invention present four principal advantages in managing thecredit risk inherent in leveraged transactions. First, a preferred formof DBAR contingent claim entails limited liability investing. Investmentliability is limited in the sense that the maximum amount a trader canlose is the amount invested. In this respect, the limited liabilityfeature is similar to that of a long option position in the traditionalmarkets. By contrast, a short option position in traditional marketsrepresents a potentially unlimited liability investment since thedownside exposure can readily exceed the option premium and is, intheory, unbounded. Importantly, a group of DBAR contingent claims of thepresent invention can easily replicate returns of a traditional shortoption position while maintaining limited liability. The limitedliability feature of a group of DBAR contingent claims is a directconsequence of the demand-side nature of the market. More specifically,in preferred embodiments there are no sales or short positions as thereare in the traditional markets, even though traders in a group of DBARcontingent claims may be able to attain the return profiles oftraditional short positions.

Second, in preferred embodiments, a trader within a group of DBARcontingent claims should have a portfolio of counterparties as describedabove. As a consequence, there should be a statistical diversificationof the credit risk such that the amount of credit risk borne by any onetrader is, on average (and in all but exceptionally rare cases), lessthan if there were an exposure to a single counterparty as is frequentlythe case in traditional markets. In other words, in preferredembodiments of the system and methods of the present invention, eachtrader is able to take advantage of the diversification effect which iswell known in portfolio analysis.

Third, in preferred embodiments of the present invention, the entiredistribution of margin loans, and the aggregate amount of leverage andcredit risk existing for a group of DBAR contingent claims, can bereadily calculated and displayed to traders at any time before thefulfillment of all of the termination criteria for the group of claims.Thus, traders themselves may have access to important informationregarding credit risk. In traditional markets such information is notreadily available.

Fourth, preferred embodiments of a DBAR contingent claim exchangeprovide more information about the distribution of possible outcomesthan do traditional market exchanges. Thus, as a byproduct of DBARcontingent claim trading according to preferred embodiments, tradershave more information about the distribution of future possible outcomesfor real-world events, which they can use to manage risk moreeffectively. For many traders, a significant part of credit risk islikely to be caused by market risk. Thus, in preferred embodiments ofthe present invention, the ability through an exchange or otherwise tocontrol or at least provide information about market risk should havepositive feedback effects for the management of credit risk.

A simple example of a group of DBAR contingent claims with the followingassumptions, illustrates these features. The example uses the followingbasic assumptions:

-   -   two defined states (with predetermined termination        criteria): (i) stock price appreciates in one month; (ii) stock        price depreciates in one month; and    -   $100 has been invested in the appreciate state, and $95 in the        depreciate state.

If a trader then invests $1 in the appreciate state, if the stock infact appreciates in the month, then the trader will be allocated apayout of $1.9406 (=196/101)—a return of $0.9406 plus the original $1investment (ignoring, for the purpose of simplicity, a transaction fee).If, before the close of the trading period the trader desireseffectively to “sell” his investment in the appreciate state, he has twochoices. He could sell the investment to a third party, which wouldnecessitate crossing of a bid and an offer in a two-way order crossingnetwork. Or, in a preferred embodiment of the method of the presentinvention, the trader can invest in the depreciate state, in proportionto the amount that had been invested in that state not counting thetrader's “new” investments. In this example, in order to fully hedge hisinvestment in the appreciate state, the trader can invest $0.95 (95/100)in the depreciate state. Under either possible outcome, therefore, thetrader will receive a payout of $1.95, i.e., if the stock appreciatesthe trader will receive 196.95/101=$1.95 and if the stock depreciatesthe trader will receive (196.95/95.95)*0.95=$1.95.

1.3 Network Implementation

A market or exchange for groups of DBAR contingent claims marketaccording to the invention is not designed to establish acounterparty-driven or order-matched market. Buyers' bids and sellers'offers do not need to be “crossed.” As a consequence of the absence of aneed for an order crossing network, preferred embodiments of the presentinvention are particularly amenable to large-scale electronic networkimplementation on a wide area network or the public Internet, forexample.

Preferred embodiments of an electronic network-based embodiment of themethod of trading in accordance with the invention include one or moreof the following features.

-   -   (a) User Accounts: DBAR contingent claims investment accounts        are established using electronic methods.    -   (b) Interest and Margin Accounts: Trader accounts are maintained        using electronic methods to record interest paid to traders on        open DBAR contingent claim balances and to debit trader balances        for margin loan interest. Interest is typically paid on        outstanding investment balances for a group of DBAR contingent        claims until the fulfillment of the termination criteria.        Interest is typically charged on outstanding margin loans while        such loans are outstanding. For some contingent claims, trade        balance interest can be imputed into the closing returns of a        trading period.    -   (c) Suspense Accounts: These accounts relate specifically to        investments which have been made by traders, during trading        periods, simultaneously in multiple states for the same event.        Multi-state trades are those in which amounts are invested over        a range of states so that, if any of the states occurs, a return        is allocated to the trader based on the closing return for the        state which in fact occurred.        -   A trader can, of course, simply break-up or divide the            multi-state investment into many separate, single-state            investments, although this approach might require the trader            to keep rebalancing his portfolio of single state            investments as returns adjust throughout the trading period            as amounts invested in each state change.        -   Multi-state trades can be used in order to replicate any            arbitrary distribution of payouts that a trader may desire.            For example, a trader might want to invest in all states in            excess of a given value or price for a security underlying a            contingent claim, e.g., the occurrence that a given stock            price exceeds 100 at some future date. The trader might also            want to receive an identical payout no matter what state            occurs among those states. For a group of DBAR contingent            claims there may well be many states for outcomes in which            the stock price exceeds 100 (e.g., greater than 100 and less            than or equal to 101; greater than 101 and less than or            equal to 102, etc.). In order to replicate a multi-state            investment using single state investments, a trader would            need continually to rebalance the portfolio of single-state            investments so that the amount invested in the selected            multi-states is divided among the states in proportion to            the existing amount invested in those states. Suspense            accounts can be employed so that the exchange, rather than            the trader, is responsible for rebalancing the portfolio of            single-state investments so that, at the end of the trading            period, the amount of the multi-state investment is            allocated among the constituent states in such a way so as            to replicate the trader's desired distribution of payouts.            Example 3.1.2 below illustrates the use of suspense accounts            for multi-state investments.    -   (d) Authentication: Each trader may have an account may be        authenticated using authenticating data.    -   (e) Data Security: The security of contingent claims        transactions over the network may be ensured, using for example        strong forms of public and private key encryption.    -   (f) Real-Time Market Data Server: Real-time market data may be        provided to support frequent calculation of returns and to        ascertain the outcomes during the observation periods.    -   (g) Real-Time Calculation Engine Server: Frequent calculation of        market returns may increase the efficient functioning of the        market. Data on coupons, dividends, market interest rates, spot        prices, and other market data can be used to calculate opening        returns at the beginning of a trading period and to ascertain        observable events during the observation period. Sophisticated        simulation methods may be required for some groups of DBAR        contingent claims in order to estimate expected returns, at        least at the start of a trading period.    -   (h) Real-Time Risk Management Server: In order to compute trader        margin requirements, expected returns for each trader should be        computed frequently. Calculations of “value-at-risk” in        traditional markets can involve onerous matrix calculations and        Monte Carlo simulations. Risk calculations in preferred        embodiments of the present invention are simpler, due to the        existence of information on the expected returns for each state.        Such information is typically unavailable in traditional capital        and reinsurance markets.    -   (i) Market Data Storage: A DBAR contingent claims exchange in        accordance with the invention may generate valuable data as a        byproduct of its operation. These data are not readily available        in traditional capital or insurance markets. In a preferred        embodiment of the present invention, investments may be        solicited over ranges of outcomes for market events, such as the        event that the 30-year U.S. Treasury bond will close on a given        date with a yield between 6.10% and 6.20%. Investment in the        entire distribution of states generates data which reflect the        expectations of traders over the entire distribution of possible        outcomes. The network implementation disclosed in this        specification may be used to capture, store and retrieve these        data.    -   (j) Market Evaluation Server: Preferred embodiments of the        method of the present invention include the ability to improve        the market's efficiency on an ongoing basis. This may readily be        accomplished, for example, by comparing the predicted returns on        a group of DBAR contingent claims returns with actual realized        outcomes. If investors have rational expectations, then DBAR        contingent claim returns will, on average, reflect trader        expectations, and these expectations will themselves be realized        on average. In preferred embodiments, efficiency measurements        are made on defined states and investments over the entire        distribution of possible outcomes, which can then be used for        statistical time series analysis with realized outcomes. The        network implementation of the present invention may therefore        include analytic servers to perform these analyses for the        purpose of continually improving the efficiency of the market.

2 FEATURES OF DBAR CONTINGENT CLAIMS

In a preferred embodiment, a group of a DBAR contingent claims relatedto an observable event includes one or more of the following features:

-   -   (1) A defined set of collectively exhaustive states representing        possible real-world outcomes related to an observable event. In        preferred embodiments, the events are events of economic        significance. The possible outcomes can typically be units of        measurement associated with the event, e.g., an event of        economic interest can be the closing index level of the S&P 500        one month in the future, and the possible outcomes can be entire        range of index levels that are possible in one month. In a        preferred embodiment, the states are defined to correspond to        one or more of the possible outcomes over the entire range of        possible outcomes, so that defined states for an event form a        countable and discrete number of ranges of possible outcomes,        and are collectively exhaustive in the sense of spanning the        entire range of possible outcomes. For example, in a preferred        embodiment, possible outcomes for the S&P 500 can range from        greater than 0 to infinity (theoretically), and a defined state        could be those index values greater than 1000 and less than or        equal to 1100. In such preferred embodiments, exactly one state        occurs when the outcome of the relevant event becomes known.    -   (2) The ability for traders to place trades on the designated        states during one or more trading periods for each event. In a        preferred embodiment, a DBAR contingent claim group defines the        acceptable units of trade or value for the respective claim.        Such units may be dollars, barrels of oil, number of shares of        stock, or any other unit or combination of units accepted by        traders and the exchange for value.    -   (3) An accepted determination of the outcome of the event for        determining which state or states have occurred. In a preferred        embodiment, a group of DBAR contingent claims defines the means        by which the outcome of the relevant events is determined. For        example, the level that the S&P 500 Index actually closed on a        predetermined date would be an outcome observation which would        enable the determination of the occurrence of one of the defined        states. A closing value of 1050 on that date, for instance,        would allow the determination that the state between 1000 and        1100 occurred.    -   (4) The specification of a DRF which takes the traded amount for        each trader for each state across the distribution of states as        that distribution exists at the end of each trading period and        calculates payouts for each investments in each state        conditioned upon the occurrence of each state. In preferred        embodiments, this is done so that the total amount of payouts        does not exceed the total amount invested by all the traders in        all the states. The DRF can be used to show payouts should each        state occur during the trading period, thereby providing to        traders information as to the collective level of interest of        all traders in each state.    -   (5) Payouts to traders for successful investments based on the        total amount of the unsuccessful investments after deduction of        the transaction fee and after fulfillment of the termination        criteria.

The states corresponding to the range of possible event outcomes arereferred to as the “distribution” or “distribution of states.” Each DBARcontingent claim group is typically associated with one distribution ofstates. The distribution will typically be defined for events ofeconomic interest for investment by traders having the expectation of areturn or a reduction of risk (“hedging”). For example, the distributioncan be based upon the values of stocks, bonds, futures, and foreignexchange rates. It can also be based upon the values of commodityindices, economic statistics (e.g., consumer price inflation monthlyreports), property-casualty losses, weather patterns for a certaingeographical region, and any other measurable or observable occurrenceor any other event in which traders would not be economicallyindifferent even in the absence of a trade on the outcome of the event.

2.1 DBAR Claim Notification

In order to facilitate further description of DBAR contingent claims,the following notation will be used in this specification:

-   -   m represents the number of traders for a given group of DBAR        contingent claims    -   n represents the number of states for a given distribution        associated with a given group of DBAR contingent claims    -   A represents a matrix with m rows and n columns, where the        element at the i-th row and j-th column, α_(ij), is the amount        that trader i has invested in state j in the expectation of a        return should state j occur    -   Π represents a matrix with n rows and n columns where element        π_(ij) is the payout per unit of investment in state i should        state j occur (“unit payouts”)    -   R represents a matrix with n rows and n columns where element        r_(ij) is the return per unit of investment in state i should        state j occur, i.e., r_(ij)=π_(ij)−1 (“unit returns”)    -   P represents a matrix with m rows and n columns, where the        element at the i-th row and j-th column, p_(ij), is the payout        to be made to trader i should state j occur, i.e., P is equal to        the matrix product A*Π.    -   P*_(j,) represents the j-th column of P, for j=1 . . . n, which        contains the payouts to each investment should state j occur    -   P_(i,)* represents the i-th row of P, for i=1 . . . m, which        contains the payouts to trader i    -   S_(i) where i=1..n, represents a state representing a range of        possible outcomes of an observable event.    -   T_(i where i=)1 . . . n, represents the total amount traded in        the expectation of the occurrence of state i    -   T represents the total traded amount over the entire        distribution of states, i.e.,

$T = {\sum\limits_{i = {1\ldots\; n}}T_{i}}$

-   -   f(A,X) represents the exchange's transaction fee, which can        depend on the entire distribution of traded amounts placed        across all the states as well as other factors, X, some of which        are identified below. For reasons of brevity, for the remainder        of this specification unless otherwise stated, the transaction        fee is assumed to be a fixed percentage of the total amount        traded over all the states.    -   c_(p) represents the interest rate charged on margin loans.    -   c_(r) represents the interest rate paid on trade balances.    -   t represents time from the acceptance of a trade or investment        to the fulfillment of all of the termination criteria for the        group of DBAR contingent claims, typically expressed in years or        fractions thereof.    -   X represents other information upon which the DRF or transaction        fee can depend such as information specific to a investment or a        trader, including for example the time or size of a trade.

In preferred embodiments, a DRF is a function which takes the tradedamounts over the distribution of states for a given group of DBARcontingent claims, the transaction fee schedule, and, conditional uponthe occurrence of each state, computes the payouts to each trade orinvestment placed over the distribution of states. In notation, such aDRF is:P=DRF(A,f(A,X),X|s=s _(i))=A* Π(A,f(A,X),X)   (DRF)

In other words, the m traders who have placed trades across the nstates, as represented in matrix A, will receive payouts as representedin matrix P should state i occur, also, taking into account thetransaction fee f and other factors X. The payouts identified in matrixP can be represented as the product of (a) the payouts per unit tradedfor each state should each state occur, as identified in the matrix Π,and (b) the matrix A which identifies the amounts traded or invested byeach trader in each state. The following notation may be used toindicate that, in preferred embodiments, payouts should not exceed thetotal amounts invested less the transaction fee, irrespective of whichstate occurs:1_(m) ^(T)*P _(*,j)+ƒ(A,X)<=1_(m) ^(T)*A*1_(n) for j=1. . . n   (DRFConstraint)where the 1 represents a column vector with dimension indicated by thesubscript, the superscript T represents the standard transpose operatorand P_(*j) is the j-th column of the matrix P representing the payoutsto be made to each trader should state j occur. Thus, in preferredembodiments, the unsuccessful investments finance the successfulinvestments. In addition, absent credit-related risks discussed below,in such embodiments there is no risk that payouts will exceed the totalamount invested in the distribution of states, no matter what stateoccurs. In short, a preferred embodiment of a group of DBAR contingentclaims of the present invention is self-financing in the sense that forany state, the payouts plus the transaction fee do not exceed the inputs(i.e., the invested amounts).

The DRF may depend on factors other than the amount of the investmentand the state in which the investment was made. For example, a payoutmay depend upon the magnitude of a change in the observed outcome for anunderlying event between two dates (e.g., the change in price of asecurity between two dates). As another example, the DRF may allocatehigher payouts to traders who initiated investments earlier in thetrading period than traders who invested later in the trading period,thereby providing incentives for liquidity earlier in the tradingperiod. Alternatively, the DRF may allocate higher payouts to largeramounts invested in a given state than to smaller amounts invested forthat state, thereby providing another liquidity incentive.

In any event, there are many possible functional forms for a DRF thatcould be used. To illustrate, one trivial form of a DRF is the case inwhich the traded amounts, A, are not reallocated at all upon theoccurrence of any state, i.e., each trader receives his traded amountback in the event that any state occurs, as indicated by the followingnotation:P=A if s=s _(i), for i=1 . . . nThis trivial DRF is not useful in allocating and exchanging risk amonghedgers.

For a meaningful risk exchange to occur, a preferred embodiment of a DRFshould effect a meaningful reallocation of amounts invested across thedistribution of states upon the occurrence of at least one state. Groupsof DBAR contingent claims of the present invention are discussed in thecontext of a canonical DRF, which is a preferred embodiment in which theamounts invested in states which did not occur are completelyreallocated to the state which did occur (less any transaction fee).This invention is not limited to a canonical DRF, and many other typesof DRFs can be used and may be preferred to implement a group of DBARcontingent claims. For example, another DRF preferred embodimentallocates half the total amount invested to the outcome state andrebates the remainder of the total amount invested to the states whichdid not occur. In another preferred embodiment, a DRF would allocatesome percentage to an occurring state, and some other percentage to oneor more “nearby” or “adjacent” states with the bulk of the non-occurringstates receiving zero payouts. Other DRFs will be apparent to those ofskill in the art from review of this specification and practice of thepresent invention.

2.2 Units of Investments and Payouts

The units of investments and payouts in systems and methods of thepresent invention may be units of currency, quantities of commodities,numbers of shares of common stock, amount of a swap transaction or anyother units representing economic value. Thus, there is no limitationthat the investments or payouts be in units of currency or money (e.g.,U.S. dollars) or that the payouts resulting from the DRF be in the sameunits as the investments. Preferably, the same unit of value is used torepresent the value of each investment, the total amount of allinvestments in a group of DBAR contingent claims, and the amountsinvested in each state.

It is possible, for example, for traders to make investments in a groupof DBAR contingent claims in numbers of shares of common stock and forthe applicable DRF to allocate payouts to traders in Japanese Yen orbarrels of oil. Furthermore, it is possible for traded amounts andpayouts to be some combination of units, such as, for example, acombination of commodities, currencies, and number of shares. Inpreferred embodiments traders need not physically deposit or receivedelivery of the value units, and can rely upon the DBAR contingent claimexchange to convert between units for the purposes of facilitatingefficient trading and payout transactions. For example, a DBARcontingent claim might be defined in such a way so that investments andpayouts are to be made in ounces of gold. A trader can still depositcurrency, e.g., U.S. dollars, with the exchange and the exchange can beresponsible for converting the amount invested in dollars into thecorrect units, e.g., gold, for the purposes of investing in a givenstate or receiving a payout. In this specification, a U.S. dollar istypically used as the unit of value for investments and payouts. Thisinvention is not limited to investments or payouts in that value unit.In situations where investments and payouts are made in different unitsor combinations of units, for purpose of allocating returns to eachinvestment the exchange preferably converts the amount of eachinvestment, and thus the total of the investments in a group of DBARcontingent claims, into a single unit of value (e.g., dollars). Example3.1.20 below illustrates a group of DBAR contingent claims in whichinvestments and payouts are in units of quantities of common stockshares.

2.3 Canonical Demand Reallocation Function

A preferred embodiment of a DRF that can be used to implement a group ofDBAR contingent claims is termed a “canonical” DRF. A canonical DRF is atype of DRF which has the following property: upon the occurrence of agiven state i, investors who have invested in that state receive apayout per unit invested equal to (a) the total amount traded for allthe states less the transaction fee, divided by (b) the total amountinvested in the occurring state. A canonical DRF may employ atransaction fee which may be a fixed percentage of the total amounttraded, T, although other transaction fees are possible. Traders whomade investments in states which not did occur receive zero payout.Using the notation developed above:

$\pi_{i,l} = \frac{\left( {1 - f} \right)*T}{T_{i}}$ if i = j, i.e., theunit payout to an investment in state iif state i occurs π_(i,l) = 0otherwise, i.e., if i ≠ j, so that the payout is zero to investments instate i if state j occurs.In a preferred embodiment of a canonical DRF, the unit payout matrix Πasdefined above is therefore a diagonal matrix with entries equal toπ_(ij) for i=j along the diagonal, and zeroes for all off-diagonalentries. For example, in a preferred embodiment for n=5 states, the unitpayout matrix is:

$\Pi = {{\left\lbrack \begin{matrix}\frac{T}{T_{1}} & 0 & 0 & 0 & 0 \\0 & \frac{T}{T_{2}} & 0 & 0 & 0 \\0 & 0 & \frac{T}{T_{3}} & 0 & 0 \\0 & 0 & 0 & \frac{T}{T_{4}} & 0 \\0 & 0 & 0 & 0 & \frac{T}{T_{5}}\end{matrix} \right\rbrack*\left( {1 - f} \right)} = {\left\lbrack \begin{matrix}\frac{1}{T_{1}} & 0 & 0 & 0 & 0 \\0 & \frac{1}{T_{2}} & 0 & 0 & 0 \\0 & 0 & \frac{1}{T_{3}} & 0 & 0 \\0 & 0 & 0 & \frac{1}{T_{4}} & 0 \\0 & 0 & 0 & 0 & \frac{1}{T_{5}}\end{matrix} \right\rbrack*T*\left( {1 - f} \right)}}$For this embodiment of a canonical DRF, the payout matrix is the totalamount invested less the transaction fee, multiplied by a diagonalmatrix which contains the inverse of the total amount invested in eachstate along the diagonal, respectively, and zeros elsewhere. Both T, thetotal amount invested by all m traders across all n states, and T_(i),the total amount invested in state i, are both functions of the matrixA, which contains the amount each trader has invested in each state:T _(i)=1_(m) ^(T)*A*B _(n)(i)T=1_(m) ^(T) *A*1_(n)where B_(n)(i) is a column vector of dimension n which has a 1 at thei-th row and zeros elsewhere. Thus, with n=5 as an example, thecanonical DRF described above has a unit payout matrix which is afunction of the amounts trade d across the states and the transactionfee:

$\Pi = {\left\lbrack \begin{matrix}\frac{1}{1_{m}^{T}*A*{B_{n}(1)}} & 0 & 0 & 0 & 0 \\0 & \frac{1}{1_{m}^{T}*A*{B_{n}(2)}} & 0 & 0 & 0 \\0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(3)}} & 0 & 0 \\0 & 0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(4)}} & 0 \\0 & 0 & 0 & 0 & \frac{1}{1_{m}^{T}*A*{B_{n}(5)}}\end{matrix} \right\rbrack*1_{m}^{T}*A*1_{n}*\left( {1 - f} \right)}$which can be generalized for any arbitrary number of states. The actualpayout matrix, in the defined units of value for the group of DBARcontingent claims (e.g., dollars), is the product of the m×n tradedamount matrix A and the n×n unit payout matrix Π, as defined above:P=A*Π(A,ƒ)   (CDRF)This provides that the payout matrix as defined above is the matrixproduct of the amounts traded as contained in the matrix A and the unitpayout matrix Π, which is itself a function of the matrix A and thetransaction fee, f. The expression is labeled CDRF for “Canonical DemandReallocation Function.”

Note that, in this preferred embodiment, any change to the matrix A willgenerally have an effect on any given trader's payout, both due tochanges in the amount invested, i.e., a direct effect through the matrixA in the CDRF, and changes in the unit payouts, i.e., an indirect effectsince the unit payout matrix Π is itself a function of the traded amountmatrix A.

2.4 Computing Investment Amounts to Achieve Desired Payouts

In preferred embodiments of a group of DBAR contingent claims of thepresent invention, some traders make investments in states during thetrading period in the expectation of a payout upon the occurrence of agiven state, as expressed in the CDRF above. Alternatively, a trader mayhave a preference for a desired payout distribution should a given stateoccur. Such a payout distribution could be denoted P_(i,*), which is arow corresponding to trader i in payout matrix P. Such a trader may wantto know how much to invest in contingent claims corresponding to a givenstate or states in order to achieve this payout distribution. In apreferred embodiment, the amount or amounts to be invested across thedistribution of states for the CDRF, given a payout distribution, can beobtained by inverting the expression for the CDRF and solving for thetraded amount matrix A:A=P*Π(A,ƒ)⁻¹   (CDRF 2)In this notation, the −1 superscript on the unit payout matrix denotes amatrix inverse.

Expression CDRF 2 does not provide an explicit solution for the tradedamount matrix A, since the unit payout matrix Π is itself a function ofthe traded amount matrix. CDRF 2 typically involves the use of numericalmethods to solve m simultaneous quadratic equations. For example,consider a trader who would like to know what amount, α, should betraded for a given state i in order to achieve a desired payout of p.Using the “forward” expression to compute payouts from traded amounts asin CDRF above yields the following equation:

$p = {\left( \frac{T + \alpha}{T_{i} + \alpha} \right)*\alpha}$This represents a given row and column of the matrix equation CDRF afterα has been traded for state i (assuming no transaction fee). Thisexpression is quadratic in the traded amount α, and can be solved forthe positive quadratic root as follows:

$\begin{matrix}{\alpha = \frac{\left( {p - T} \right) + \sqrt{\left( {p - T} \right)^{2} + {4*p*T_{i}}}}{2}} & \left( {{CDRF}\mspace{14mu} 3} \right)\end{matrix}$

2.5 A Canonical DRF Example

A simplified example illustrates the use of the CDRF with a group ofDBAR contingent claims defined over two states (e.g., states “1” and“2”) in which four traders make investments. For the example, thefollowing assumptions are made: (1) the transaction fee, f, is zero; (2)the investment and payout units are both dollars; (3) trader 1 has madeinvestments in the amount of $5 in state 1 and $10 state 2; and (4)trader 2 has made an investment in the amount of $7 for state 1 only.With the investment activity so far described, the traded amount matrixA, which as 4 rows and 2 columns, and the unit payout matrix Π which has2 rows and 2 columns, would be denoted as follows:

$A = \begin{matrix}5 & 10 \\7 & 0 \\0 & 0 \\0 & 0\end{matrix}$ $\Pi = {\begin{bmatrix}\frac{1}{12} & 0 \\0 & \frac{1}{10}\end{bmatrix}*22}$

The payout matrix P, which contains the payouts in dollars for eachtrader should each state occur is, the product of A and Π:

$P = \begin{matrix}{\mspace{14mu} 9.167} & 22 \\12.833 & 0 \\{\mspace{14mu} 0} & 0 \\{\mspace{14mu} 0} & 0\end{matrix}$The first row of P corresponds to payouts to trader 1 based on hisinvestments and the unit payout matrix. Should state 1 occur, trader 1will receive a payout of $9.167 and will receive $22 should state 2occur. Similarly, trader 2 will receive $12.833 should state 1 occur and$0 should state 2 occur (since trader 2 did not make any investment instate 2). In this illustration, traders 3 and 4 have $0 payouts sincethey have made no investments.

In accordance with the expression above labeled “DRF Constraint,” thetotal payouts to be made upon the occurrence of either state is lessthan or equal to the total amounts invested. In other words, the CDRF inthis example is self-financing so that total payouts plus thetransaction fee (assumed to be zero in this example) do not exceed thetotal amounts invested, irrespective of which state occurs. This isindicated by the following notation:1_(m) ^(T) *P _(*,1)=22≦1_(m) ^(T) *A*1_(n)=221_(m) ^(T) *P _(*,2)=22≦1_(m) ^(T) *A*1_(n)=22

Continuing with this example, it is now assumed that traders 3 and 4each would like to make investments which generate a desired payoutdistribution. For example, it is assumed that trader 3 would like toreceive a payout of $2 should state 1 occur and $4 should state 2 occur,while trader 4 would like to receive a payout of $5 should state 1 occurand $0 should state 2 occur. In the CDRF notation:P _(3,*)=[2 4]P _(4,*)=[5 0]

In a preferred embodiment and this example, payouts are made based uponthe invested amounts A, and therefore are also based on the unit payoutmatrix Π(A,f(A)), given the distribution of traded amounts as they existat the end of the trading period. For purposes of this example, it isnow further assumed (a) that at the end of the trading period traders 1and 2 have made investments as indicated above, and (b) that the desiredpayout distributions for traders 3 and 4 have been recorded in asuspense account which is used to determine the allocation ofmulti-state investments to each state in order to achieve the desiredpayout distributions for each trader, given the investments by the othertraders as they exist at the end of the trading period. In order todetermine the proper allocation, the suspense account can be used tosolve CDRF 2 for example:

$\begin{bmatrix}5 & 10 \\7 & {\mspace{14mu} 0} \\\alpha_{3,1} & \alpha_{3,2} \\\alpha_{4,1} & \alpha_{4,2}\end{bmatrix} = {\begin{bmatrix}p_{1,1} & p_{1,2} \\p_{2,1} & p_{2,2} \\2 & 4 \\5 & 0\end{bmatrix}*\mspace{40mu}\left\lbrack \begin{matrix}\frac{1}{\left( {5 + 7 + \alpha_{3,1} + \alpha_{4,1}} \right)} & 0 \\0 & \frac{1}{\left( {10 + 0 + \alpha_{3,2} + \alpha_{4,2}} \right)}\end{matrix} \right\rbrack*\left( {5 + 10 + 7 + 0 + \alpha_{3,1} + \alpha_{4,1} + \alpha_{3,2} + \alpha_{4,2}} \right)}$The solution of this expression will yield the amounts that traders 3and 4 need to invest in for contingent claims corresponding to states 1and 2 to in order to achieve their desired payout distributions,respectively. This solution will also finalize the total investmentamount so that traders 1 and 2 will be able to determine their payoutsshould either state occur. This solution can be achieved using acomputer program which computes an investment amount for each state foreach trader in order to generate the desired payout for that trader forthat state. In a preferred embodiment, the computer program repeats theprocess iteratively until the investment amounts calculated converge,i.e., so that the amounts to be invested by traders 3 and 4 no longermaterially change with each successive iteration of the computationalprocess. This method is known in the art as fixed point iteration and isexplained in more detail in the Technical Appendix. The following tablecontains a computer code listing of two functions written in Microsoft'sVisual Basic which can be used to perform the iterative calculations tocompute the final allocations of the invested amounts in this example ofa group of DBAR contingent claims with a Canonical Demand ReallocationFunction:

TABLE 1 Illustrative Visual Basic Computer Code for Solving CDRF 2Function allocatetrades(A_mat, P_mat) As Variant DimA_final Dim tradesAs Long Dim states As Long trades = P_mat.Rows.Count states =P_mat.Columns.Count ReDim A_final(1 To trades, 1 To states) ReDimstatedem(1 To states) Dim i As Long Dim totaldemand As Double Dim totaldesired As Double Dim iterations As Long iterations = 10 For i = 1 Totrades  For j = 1 To states  statedem(j) = A_mat(i, j) + statedem(j) A_final(i, j) = A_mat(i, j)  Next j Next i For i = 1 To states totaldemand = totaldemand + statedem(i) Next i For i = 1 To iterations For j = 1 To trades  For z = 1 To states   If A_mat(j, z) = 0 Then  totaldemand = totaldemand − A_final(j, z)   statedem(z) = statedem(z)− A_final(j, z)   tempalloc = A_final(j, z)   A_final(j, z) =stateall(totaldemand, P_mat(j, z), statedem(z))   totaldemand =A_final(j, z) + totaldemand   statedem(z) = A_final(j, z) + statedem(z) End If  Next z  Next j Next i allocatetrades = A_final End FunctionFunction stateall(totdemex, despaystate, totstateex)  Dim sol1 As Double sol1 = (−(totdemex − despaystate) + ((totdemex − despaystate){circumflex over ( )} 2 +   4 * despaystate * totstateex) {circumflexover ( )} 0.5)/2  stateall = sol1 End FunctionFor this example involving two states and four traders, use of thecomputer code represented in Table 1 produces an investment amountmatrix A, as follows:

$A = \begin{matrix}5 & 10 \\7 & {\mspace{14mu} 0} \\1.1574 & {\mspace{14mu} 1.6852} \\2.8935 & {\mspace{14mu} 0}\end{matrix}$The matrix of unit payouts, Π, can be computed from A as described aboveand is equal to:

$\Pi = \begin{matrix}1.728 & 0 \\0 & 2.3736\end{matrix}$The resulting payout matrix P is the product of A and Π and is equal to:

$P = \begin{matrix}{\mspace{14mu} 8.64} & 23.7361 \\12.0961 & {\mspace{14mu} 0} \\{\mspace{14mu} 2} & {\mspace{14mu} 4} \\{\mspace{14mu} 5} & {\mspace{14mu} 0}\end{matrix}$It can be noted that the sum of each column of P above is equal to27.7361, which is equal (in dollars) to the total amount invested so, asdesired in this example, the group of DBAR contingent claims isself-financing. The allocation is said to be in equilibrium, since theamounts invested by traders 1 and 2 are undisturbed, and traders 3 and 4receive their desired payouts, as specified above, should each stateoccur.

2.6 Interest Considerations

When investing in a group of DBAR contingent claims, traders willtypically have outstanding balances invested for periods of time and mayalso have outstanding loans or margin balances from the exchange forperiods of time. Traders will typically be paid interest on outstandinginvestment balances and typically will pay interest on outstandingmargin loans. In preferred embodiments, the effect of trade balanceinterest and margin loan interest can be made explicit in the payouts,although in alternate preferred embodiments these items can be handledoutside of the payout structure, for example, by debiting and creditinguser accounts. So, if a fraction β of a trade of one value unit is madewith cash and the rest on margin, the unit payout π_(i) in the eventthat state i occurs can be expressed as follows:

$\pi_{i} = {\frac{\left( {1 - f} \right)*T}{T_{i}} + {\beta*\left( c_{r} \right)*t_{b}} - {\left( {1 - \beta} \right)*\left( c_{p} \right)*t_{l}}}$where the last two terms express the respective credit for tradebalances per unit invested for time t_(b) and debit for margin loans perunit invested for time t₁.

2.7 Returns and Probabilities

In a preferred embodiment of a group of DBAR contingent claims with acanonical DRF, returns which represent the percentage return per unit ofinvestment are closely related to payouts. Such returns are also closelyrelated to the notion of a financial return familiar to investors. Forexample, if an investor has purchased a stock for $100 and sells it for$110, then this investor has realized a return of 10% (and a payout of$110).

In a preferred embodiment of a group of DBAR contingent claims with acanonical, the unit return, r_(i), should state i occur may be expressedas follows:

$\begin{matrix}{r_{l} = \frac{{\left( {1 - f} \right)*{\sum\limits_{{i = 1},n}T_{i}}} - T_{l}}{T_{i}}} & \text{if~~state~~i~~occurs} \\{r_{i} = {- 1}} & \text{otherwise,~~i.e., if~~state~~i~~does~~not~~occur}\end{matrix}$

In such an embodiment, the return per unit investment in a state thatoccurs is a function of the amount invested in that state, the amountinvested in all the other states and the exchange fee. The unit returnis −100% for a state that does not occur, i.e., the entire amountinvested in the expectation of receiving a return if a state occurs isforfeited if that state fails to occur. A −100% return in such an eventhas the same return profile as, for example, a traditional optionexpiring “out of the money.” When a traditional option expires out ofthe money, the premium decays to zero, and the entire amount invested inthe option is lost.

For purposes of this specification, a payout is defined as one plus thereturn per unit invested in a given state multiplied by the amount thathas been invested in that state. The sum of all payouts P_(s), for agroup of DBAR contingent claims corresponding to all n possible statescan be expressed as follows:

$\begin{matrix}{{P_{s} = {{\left( {1 + r_{i}} \right)*T_{i}} + {\sum\limits_{j,{j \neq i}}{\left( {1 + r_{j}} \right)*T_{j}}}}}\;} & {i,{j = {1\mspace{14mu}\ldots\mspace{14mu} n}}}\end{matrix}$In a preferred embodiment employing a canonical DRF, the payout hd s maybe found for the occurrence of state i by substituting the aboveexpressions for the unit return in any state:

$P_{s} = {{{\left( {\frac{{\left( {1 - f} \right)*{\sum\limits_{l = {1\ldots\; n}}T_{l}}} - T_{i}}{T_{i}} + 1} \right)*T_{l}} + {\sum\limits_{j,{j \neq i}}{\left( {{- 1} + 1} \right)*T_{j}}}} = {\left( {1 - f} \right)*{\sum\limits_{i = {l\ldots n}}T_{i}}}}$

Accordingly, in such a preferred embodiment, for the occurrence of anygiven state, no matter what state, the aggregate payout to all of thetraders as a whole is one minus the transaction fee paid to the exchange(expressed in this preferred embodiment as a percentage of totalinvestment across all the states), multiplied by the total amountinvested across all the states for the group of DBAR contingent claims.This means that in a preferred embodiment of a group of the DBARcontingent claims, and assuming no credit or similar risks, to theexchange, (1) the exchange has zero probability of loss in any givenstate; (2) for the occurrence of any given state, the exchange receivesan exchange fee and is not exposed to any risk; (3) payouts and returnsare a function of demand flow, i.e., amounts invested; and (4)transaction fees or exchange fees can be a simple function of aggregateamount invested.

Other transaction fees can be implemented. For example, the transactionfee might have a fixed component for some level of aggregate amountinvested and then have either a sliding or fixed percentage applied tothe amount of the investment in excess of this level. Other methods fordetermining the transaction fee are apparent to those of skill in theart, from this specification or based on practice of the presentinvention.

In a preferred embodiment, the total distribution of amounts invested inthe various states also implies an assessment by all traderscollectively of the probabilities of occurrence of each state. In apreferred embodiment of a group of DBAR contingent claims with acanonical DRF, the expected return E(r_(i)) for an investment in a givenstate i (as opposed to the return actually received once outcomes areknown) may be expressed as the probability weighted sum of the returns:E(r _(i))=q _(i) *r _(i)+(1−q _(i))*−1=q _(i)*(1+r ₁)−1Where q_(i) is the probability of the occurrence of state i implied bythe matrix A (which contains all of the invested amounts for all statesin the group of DBAR contingent claims). Substituting the expression forr_(i) from above yields:

${E\left( r_{i} \right)} = {{q_{i}*\left( \frac{\left( {1 - f} \right)*{\sum\limits_{l}T_{l}}}{T_{l}} \right)} - 1}$

In an efficient market, the expected return E(r_(i)) across all statesis equal to the transaction costs of trading, i.e., on average, alltraders collectively earn returns that do not exceed the costs oftrading. Thus, in an efficient market for a group of DBAR contingentclaims using a canonical, where E(r_(i)) equals the transaction fee, −f,the probability of the occurrence of state i implied by matrix A iscomputed to be:

$q_{i} = \frac{T_{l}}{\sum\limits_{l}T_{l}}$

Thus, in such a group of DBAR contingent claims, the implied probabilityof a given state is the ratio of the amount invested in that statedivided by the total amount invested in all states. This relationshipallows traders in the group of DBAR contingent claims (with a canonicalDRF) readily to calculate the implied probability which traders attachto the various states.

Information of interest to a trader typically includes the amountsinvested per state, the unit return per state, and implied stateprobabilities. An advantage of the DBAR exchange of the presentinvention is the relationship among these quantities. In a preferredembodiment, if the trader knows one, the other two can be readilydetermined. For example, the relationship of unit returns to theoccurrence of a state and the probability of the occurrence of thatstate implied by A can be expressed as follows:

$q_{l} = \frac{\left( {1 - f} \right)}{\left( {1 + r_{l}} \right)}$

The expressions derived above show that returns and implied stateprobabilities may be calculated from the distribution of the investedamounts, T_(i), for all states, i=1 . . . n. In the traditional markets,the amount traded across the distribution of states (limit order book),is not readily available. Furthermore, in traditional markets there areno such ready mathematical calculations which relate with any precisioninvested amounts or the limit order book to returns or prices whichclear the market, i.e., prices at which the supply equals the demand.Rather, in the traditional markets, specialist brokers and market makerstypically have privileged access to the distribution of bids and offers,or the limit order book, and frequently use this privileged informationin order to set market prices which balance supply and demand at anygiven time in the judgment of the market maker.

2.8 Computations When Invested Amounts Are Large

In a preferred embodiment of a group of DBAR contingent claims using acanonical DRF, when large amounts are invested across the distributionof states, it may be possible to perform approximate investmentallocation calculations in order to generate desired payoutdistributions. The payout, p, should state i occur for a trader whoconsiders making an investment of size α in state i has been shown aboveto be:

$p = {\left( \frac{T + \alpha}{T_{l} + \alpha} \right)*\alpha}$If α is small compared to both the total invested in state i and thetotal amount invested in all the states, then adding α to state i willnot have a material effect on the ratio of the total amount invested inall the states to the total amount invested in state i. In thesecircumstances,

$\frac{T + \alpha}{T_{i} + \alpha} \approx \frac{T}{T_{i}}$Thus, in preferred embodiments where an approximation is acceptable, thepayout to state i may be expressed as:

$p \approx {\frac{T}{T_{i}}*\alpha}$In these circumstances, the investment needed to generate the payout pis:

${\alpha \approx {\frac{T_{i}}{T}*p}} = {q_{i}*p}$These expressions indicate that in preferred embodiments, the amount tobe invested to generate a desired payout is approximately equal to theratio of the total amount invested in state i to the total amountinvested in all states, multiplied by the desired payout. This isequivalent to the implied probability multiplied by the desired payout.Applying this approximation to the expression CDRF 2, above, yields thefollowing:A≈P*∉ ⁻¹ =P*Qwhere the matrix Q, of dimension n x n, is equal to the inverse of unitpayouts Π, and has along the diagonals q_(i) for i=1 . . . n. Thisexpression provides an approximate but more readily calculable solutionto CDRF 2 as the expression implicitly assumes that an amount investedby a trader has approximately no effect on the existing unit payouts orimplied probabilities. This approximate solution, which is linear andnot quadratic, will sometimes be used in the following examples where itcan be assumed that the total amounts invested are large in relation toany given trader's particular investment.

3 EXAMPLES OF GROUPS OF DBAR CONTINGENT CLAIMS 3.1 DBAR RangeDerivatives

A DBAR Range Derivative (DBAR RD) is a type of group of DBAR contingentclaims implemented using a canonical DRF described above. In a DBAR RD,a range of possible outcomes associated with an observable event ofeconomic significance is partitioned into defined states. In a preferredembodiment, the states are defined as discrete ranges of possibleoutcomes so that the entire distribution of states covers all thepossible outcomes—that is, the states are collectively exhaustive.Furthermore, in a DBAR RD, states are preferably defined so as to bemutually exclusive as well, meaning that the states are defined in sucha way so that exactly one state occurs. If the states are defined to beboth mutually exclusive and collectively exhaustive, the states form thebasis of a probability distribution defined over discrete outcomeranges. Defining the states in this way has many advantages as describedbelow, including the advantage that the amount which traders investacross the states can be readily converted into implied probabilitiesrepresenting the collective assessment of traders as to the likelihoodof the occurrence of each state.

The system and methods of the present invention may also be applied todetermine projected DBAR RD returns for various states at the beginningof a trading period. Such a determination can be, but need not be, madeby an exchange. In preferred embodiments of a group of DBAR contingentclaims the distribution of invested amounts at the end of a tradingperiod determines the returns for each state, and the amount invested ineach state is a function of trader preferences and probabilityassessments of each state. Accordingly, some assumptions typically needto be made in order to determine preliminary or projected returns foreach state at the beginning of a trading period.

An illustration is provided to explain further the operation of DBARRDs. In the following illustration, it is assumed that all traders arerisk neutral so that implied probabilities for a state are equal to theactual probabilities, and so that all traders have identical probabilityassessments of the possible outcomes for the event defining thecontingent claim. For convenience in this illustration, the eventforming the basis for the contingent claims is taken to be a closingprice of a security, such as a common stock, at some future date; andthe states, which represent the possible outcomes of the level of theclosing price, are defined to be distinct, mutually exclusive andcollectively exhaustive of the range of (possible) closing prices forthe security. In this illustration, the following notation is used:

-   -   τ represents a given time during the trading period at which        traders are making investment decisions    -   θ represents the time corresponding to the expiration of the        contingent claim    -   V_(τ) represents the price of underlying security at time τ    -   V_(θ) represents the price of underlying security at time θ    -   Z(τ,θ) represents the present value of one unit of value payable        at time θ evaluated at time τ    -   D(τ,θ) represents dividends or coupons payable between time τ        and θ    -   σ represents annualized volatility of natural logarithm returns        of the underlying security    -   dz represents the standard normal variate        Traders make choices at a representative time, τ, during a        trading period which is open, so that time τ is temporally        subsequent to the current trading period's TSD.

In this illustration, and in preferred embodiments, the defined statesfor the group of contingent claims for the fmal closing price V_(θ) areconstructed by discretizing the full range of possible prices intopossible mutually exclusive and collectively exhaustive states. Thetechnique is similar to forming a histogram for discrete countable data.The endpoints of each state can be chosen, for example, to be equallyspaced, or of varying spacing to reflect the reduced likehood of extremeoutcomes compared to outcomes near the mean or median of thedistribution. States may also be defined in other manners apparent toone of skill in the art. The lower endpoint of a state can be includedand the upper endpoint excluded, or vice versa. In any event, inpreferred embodiments, the states are defined (as explained below) tomaximize the attractiveness of investment in the group of DBARcontingent claims, since it is the invested amounts that ultimatelydetermine the returns that are associated with each defined state.

The procedure of defining states, for example for a stock price, can beaccomplished by assuming lognormality, by using statistical estimationtechniques based on historical time series data and cross-section marketdata from options prices, by using other statistical distributions, oraccording to other procedures known to one of skill in the art orlearned from this specification or through practice of the presentinvention. For example, it is quite common among derivatives traders toestimate volatility parameters for the purpose of pricing options byusing the econometric techniques such as GARCH. Using these parametersand the known dividend or coupons over the time period from τ to θ, forexample, the states for a DBAR RD can be defined.

A lognormal distribution is chosen for this illustration since it iscommonly employed by derivatives traders as a distributional assumptionfor the purpose of evaluating the prices of options and other derivativesecurities. Accordingly, for purposes of this illustration it is assumedthat all traders agree that the underlying distribution of states forthe security are lognormally distributed such that:

${\overset{\sim}{V}}_{\theta} = {\left( {\frac{V_{\tau}}{Z\left( {\tau,\theta} \right)} - \frac{D\left( {\tau,\theta} \right)}{Z\left( {\tau,\theta} \right)}} \right)*{\mathbb{e}}^{{{- \sigma^{2}}/2}*{({\theta - \tau})}}*{\mathbb{e}}^{\sigma*\sqrt{\theta - \tau}*{dz}}}$where the “tilde” on the left-hand side of the expression indicates thatthe final closing price of the value of the security at time θ is yet tobe known. Inversion of the expression for dz and discretization ofranges yields the following expressions:

$\begin{matrix}{{dz} = {{\ln\left( \frac{V_{\theta}*{\mathbb{e}}^{\frac{\theta^{2}}{2}*{({\theta - \tau})}}}{\left( {\frac{V_{\tau}}{Z\left( {\tau,\theta} \right)} - \frac{D\left( {\tau,\theta} \right)}{Z\left( {\tau,\theta} \right)}} \right)} \right)}/\left( {\sigma*\sqrt{\theta - \tau}} \right)}} \\{{q_{l}\left( {V_{l} \leq V_{\theta} <_{l + 1}} \right)} = {{{cdf}\left( {dz}_{l + 1} \right)} - {{cdf}\left( {dz}_{i} \right)}}} \\{{r_{i}\left( {V_{l} \leq V_{\theta} <_{l + 1}} \right)} = {\frac{\left( {1 - f} \right)}{q_{l}\left( {V_{l} \leq V_{\theta} < V_{l + 1}} \right)} - 1}}\end{matrix}$where cdf(dz) is the cumulative standard normal function.

The assumptions and calculations reflected in the expressions presentedabove can also be used to calculate indicative returns (“openingreturns”), r_(i) at a beginning of the trading period for a given groupof DBAR contingent claims. In a preferred embodiment, the calculatedopening returns are based on the exchange's best estimate of theprobabilities for the states defining the claim and therefore mayprovide good indications to traders of likely returns once trading isunderway. Of course, the opening returns need not be provided at all, astraded amounts placed throughout the trading period allows thecalculation of actual expected returns at any time during the tradingperiod.

The following examples of DBAR range derivatives and other contingentclaims serve to illustrate their operation, their usefulness inconnection with events of economic significance involving inherent riskor uncertainty, the advantages of exchanges for groups of DBARcontingent claims, and, more generally, systems and methods of thepresent invention.

In each of these examples, a state is defined to include a range ofpossible outcomes of an event of economic significance (e.g., the priceof a stock). A curved brace “(” or “)” denotes strict inequality (e.g.,“greater than” or “less than,” respectively) and a square brace “]” or“[” shall denote weak inequality (e.g., “less than or equal to” or“greater than or equal to,” respectively). For simplicity, and unlessotherwise stated, the following examples also assume that the exchangetransaction fee, f, is zero.

Example 3.1.1: DBAR Contingent Claim On Underlying Common Stock

Underlying Security: Microsoft Corporation Common Stock (“MSFT”) Date:Aug. 18, 1999 Spot Price: 85 Market Volatility: 50% annualized TradingStart Date: Aug. 18, 1999, Market Open Trading End Date: Aug. 18, 1999,Market Close Expiration: Aug. 19, 1999, Market Close Event: MSFT ClosingPrice at Expiration Trading Time: 1 day Duration to TED: 1 day DividendsPayable to Expiration: 0 Interbank short-term interest rate 5.5%(Actual/360 to Expiration: daycount) Present Value factor to Expiration:0.999847 Investment and Payout Units: U.S. Dollars (“USD”)

In this Example 3.11, the predetermined termination criteria are theinvestment in a contingent claim during the trading period and theclosing of the market for Microsoft common stock on Aug. 19, 1999.

If all traders agree that the underlying distribution of closing pricesis lognormally distributed with volatility of 50%, then an illustrative“snapshot” distribution of invested amounts and returns for $100 millionof aggregate investment can be readily calculated to yield the followingtable.

TABLE 3.1.1-1 States Investment in State (′000) Return Per Unit if StateOccurs (0,80] 1,046.58 94.55 (80,80.5] 870.67 113.85 (80.5,81] 1,411.3569.85 (81,81.5] 2,157.85 45.34 (81.5,82] 3,115.03 31.1 (82,82.5]4,250.18 22.53 (82.5,83] 5,486.44 17.23 (83,83.5] 6,707.18 13.91(83.5,84] 7,772.68 11.87 (84,84.5] 8,546.50 10.7 (84.5,85] 8,924.71 10.2(85,85.5] 8,858.85 10.29 (85.5,86] 8,366.06 10.95 (86,86.5] 7,523.1312.29 (86.5,87] 6,447.26 14.51 (87,87.5] 5,270.01 17.98 (87.5,88]4,112.05 23.31 (88,88.5] 3,065.21 31.62 (88.5,89] 2,184.5 44.78(89,89.5] 1,489.58 66.13 (89.5,90] 972.56 101.82 (90,∞] 1,421.61 69.34

Consistent with the design of a preferred embodiment of a group of DBARcontingent claims, the amount invested for any given state is inverselyrelated to the unit return for that state.

In preferred embodiments of groups of DBAR contingent claims, traderscan invest in none, one or many states. It may be possible in preferredembodiments to allow traders efficiently to invest in a set, subset orcombination of states for the purposes of generating desireddistributions of payouts across the states. In particular, traders maybe interested in replicating payout distributions which are common inthe traditional markets, such as payouts corresponding to a long stockposition, a short futures position, or a long option straddle position.

If in this Example 3.1.1 a trader desired to hedge his exposure toextreme outcomes in MSFT stock, then the trader could invest in statesat each end of the distribution of possible outcomes. For instance, atrader might decide to invest $100,000 in states encompassing pricesfrom $0 up to and including $83 (i.e., (0,83]) and another $100,000 instates encompassing prices greater than $86.50 (i.e., (86.5,∞]). Thetrader may further desire that no matter what state actually occurswithin these ranges (should the state occur in either range) upon thefulfillment of the predetermined termination criteria, an identicalpayout will result. In this Example 3.1.1, a multi-state investment iseffectively a group of single state investments over each multi-staterange, where an amount is invested in each state in the range inproportion to the amount previously invested in that state. If, forexample, the returns provided in Table 3.1.1-1 represent finalizedprojected returns at the end of the trading period, then eachmulti-state investment may be allocated to its constituent states on apro-rata or proportional basis according to the relative amountsinvested in the constituent states at the close of trading. In this way,more of the multi-state investment is allocated to states with largerinvestments and less allocated to the states with smaller investments.

Other desired payout distributions across the states can be generated byallocating the amount invested among the constituent states in differentways so as achieve a trader's desired payout distribution. A trader mayselect, for example, both the magnitude of the payouts and how thosepayouts are to be distributed should each state occur and let the DBARexchange's multi-state allocation methods determine (1) the size of theamount invested in each particular constituent state; (2) the states inwhich investments will be made, and (3) how much of the total amount tobe invested will be invested in each of the states so determined. Otherexamples below demonstrate how such selections may be implemented.

Since in preferred embodiments the final projected returns are not knownuntil the end of a given trading period, in such embodiments a previousmulti-state investment is reallocated to its constituent statesperiodically as the amounts invested in each state (and thereforereturns) change during the trading period. At the end of the tradingperiod when trading ceases and projected returns are finalized, in apreferred embodiment a final reallocation is made of all the multi-stateinvestments. In preferred embodiments, a suspense account is used torecord and reallocate multi-state investments during the course oftrading and at the end of the trading period.

Referring back to the illustration assuming two multi-state trades overthe ranges (0,83] and (86.5,∞] for MSFT stock, Table 3.1.1-2 shows howthe multi-state investments in the amount of $100,000 each could beallocated according to a preferred embodiment to the individual statesover each range in order to achieve a payout for each multi-state rangewhich is identical regardless of which state occurs within each range.In particular, in this illustration the multi-state investments areallocated in proportion to the previously invested amount in each statc,and the multi-state investments marginally lower returns over (0,83] and(86.5,∞], but marginally increase returns over the range (83, 86.5] asexpected.

To show that the allocation in this example has achieved its goal ofdelivering the desired payouts to the trader, two payouts for the (0,83] range are considered. The payout if constituent state (80.5, 81]occurs is the amount invested in that state ($7.696) multiplied by oneplus the return per unit if that state occurs, or(1+69.61)*7.696=$543.40. A similar analysis for the state (82.5, 83]shows that, if it occurs, the payout is equal to(1+17.162)*29.918=$543.40. Thus, in this illustration, the traderreceives the same payout no matter which constituent state occurs withinthe multi-state investment. Similar calculations can be performed forthe range [86.5,∞]. For example, under the same assumptions, the payoutfor the constituent state [86.5,87] would receive a payout of $399.80 ifthe stock price fill in that range after the fulfillment of all of thepredetermined termination criteria. In this illustration, eachconstituent state over the range [86.5, ∞] would receive a payout of$399.80, no matter which of those states occurs.

TABLE 3.1.1-2 Traded Amount in State Return Per Unit Multi-State States(′000) if State Occurs Allocation (′000) (0,80] 1052.29 94.22 5.707(80,80.5] 875.42 113.46 4.748 (80.5,81] 1,419.05 69.61 7.696 (81,81.5]2,169.61 45.18 11.767 (81.5,82] 3,132.02 30.99 16.987 (82,82.5] 4,273.3522.45 23.177 (82.5,83] 5,516.36 17.16 29.918 (83,83.5] 6,707.18 13.94(83.5,84] 7,772.68 11.89 (84,84.5] 8,546.50 10.72 (84.5,85] 8,924.7110.23 (85,85.5] 8,858.85 10.31 (85.5,86] 8,366.06 10.98 (86,86.5]7,523.13 12.32 (86.5,87] 6,473.09 14.48 25.828 (87,87.5] 5,291.12 17.9421.111 (87.5,88] 4,128.52 23.27 16.473 (88,88.5] 3,077.49 31.56 12.279(88.5,89] 2,193.25 44.69 8.751 (89,89.5] 1,495.55 66.00 5.967 (89.5,90]976.46 101.62 3.896 (90,∞] 1,427.31 69.20 5.695

Example 3.1.2: Multiple Multi-State Investments

If numerous multi-state investments are made for a group of DBARcontingent claims, then in a preferred embodiment an iterative procedurecan be employed to allocate all of the multi-state investments to theirrespective constituent states. in preferred embodiments, the goal wouldbe to allocate each multi-state investment in response to changes inamounts invested during the trading period, and to make a finalallocation at the end of the trading period so that each multi-stateinvestment generates the payouts desired by the respective trader. Inpreferred embodiments, the process of allocating multi-state investmentscan be iterative, since allocations depend upon the amounts tradedacross the distribution of states at any point in time. As aconsequence, in preferred embodiments, a given distribution of investedamounts will result in a certain allocation of a multi-state investment.When another multi-state investment is allocated, the distribution ofinvested amounts across the defined states may change and thereforenecessitate the reallocation of any previously allocated multi-stateinvestments. In such preferred embodiments, each multi-state allocationis re-performed so that, after a number of iterations through all of thepending multi-state investments, both the amounts invested and theirallocations among constituent states in the multi-state investments nolonger change with each successive iteration and a convergence isachieved. In preferred embodiments, when convergence is achieved,further iteration and reallocation among the multi-state investments donot change any multi-state allocation, and the entire distribution ofamounts invested across the states remains stable. Computer code, asillustrated in Table 1 above or related code readily apparent to one ofskill in the art, can be used to implement this iterative procedure.

A simple example demonstrates a preferred embodiment of an iterativeprocedure that may be employed. For purposes of this example, apreferred embodiment of the following assumptions are made: (i) thereare four defined states for the group of DBAR contingent claims; (ii)prior to the allocation of any multi-state investments, $100 has beeninvested in each state so that the unit return for each of the fourstates is 3; (iii) each desires that each constituent state in amulti-state investment provides the same payout regardless of whichconstituent state actually occurs; and (iv) that the following othermulti-state investments have been made:

TABLE 3.1.2-1 Investment Invested Number State 1 State 2 State 3 State 4Amount, $ 1001 X X 0 0 100 1002 X 0 X X 50 1003 X X 0 0 120 1004 X X X 0160 1005 X X X 0 180 1006 0 0 X X 210 1007 X X X 0 80 1008 X 0 X X 9501009 X X X 0 1000 1010 X X 0 X 500 1011 X 0 0 X 250 1012 X X 0 0 1001013 X 0 X 0 500 1014 0 X 0 X 1000 1015 0 X X 0 170 1016 0 X 0 X 1201017 X 0 X 0 1000 1018 0 0 X X 200 1019 X X X 0 250 1020 X X 0 X 3001021 0 X X X 100 1022 X 0 X X 400where an “X” in each state represents a constituent state of themulti-state trade. Thus, as depicted in Table 3.1.2-1, trade number 1001in the first row is a multi-state investment of $100 to be allocatedamong constituent states 1 and 2, trade number 1002 in the second row isanother multi-state investment in the amount of $50 to be allocatedamong constituent states 1, 3, and 4; etc.

Applied to the illustrative multi-state investment described above, theiterative procedure described above and embodied in the illustrativecomputer code in Table 1, results in the following allocations:

TABLE 3.1.2-2 Investment Number State 1 ($) State 2 ($) State 3 ($)State 4 ($) 1001 73.8396 26.1604 0 0 1002 26.66782 0 12.53362 10.798561003 88.60752 31.39248 0 0 1004 87.70597 31.07308 41.22096 0 100598.66921 34.95721 46.37358 0 1006 0 0 112.8081 97.19185 1007 43.8529815.53654 20.61048 0 1008 506.6886 0 238.1387 205.1726 1009 548.1623194.2067 257.631 0 1010 284.2176 100.6946 0 115.0878 1011 177.945 0 072.055 1012 73.8396 26.1604 0 0 1013 340.1383 0 159.8617 0 1014 0466.6488 0 533.3512 1015 0 73.06859 96.93141 0 1016 0 55.99785 064.00215 1017 680.2766 0 319.7234 0 1018 0 0 107.4363 92.56367 1019137.0406 48.55168 64.40774 0 1020 170.5306 60.41675 0 69.05268 1021 028.82243 38.23529 32.94229 1022 213.3426 0 100.2689 86.38848In Table 3.1.2-2 each row shows the allocation among the constituentstates of the multi-state investment entered into the corresponding rowof Table 3.1.2-1, the first row of Table 3.1.2-2 that investment number1001 in the amount of $100 has been allocated $73.8396 to state 1 andthe remainder to state 2.

It may be shown that the multi-state allocations identified above resultin payouts to traders which are desired by the traders—that is, in thisexample the desired payouts are the same regardless of which stateoccurs among the constituent states of a given multi-state investment.Based on the total amount invested as reflected in Table 3.1.2-2 andassuming a zero transaction fee, the unit returns for each state are:

State 1 State 2 State 3 State 4 Return Per Dollar 1.2292 5.2921 3.74314.5052 InvestedConsideration of Investment 1022 in this example, illustrates theuniformity of payouts for each state in which an investment is made(i.e., states 1, 3 and 4). If state 1 occurs, the total payout to thetrader is the unit return for state 1—1.2292—multiplied by the amounttraded for state 1 in trade 1022—$213.3426—plus the initialtrade—$213.3426. This equals 1.2292*213.3426+213.3426=$475.58. If state3 occurs, the payout is equal to 3.7431*100.2689+100.2689=$475.58.Finally, if state 4 occurs, the payout is equal to4.5052*86.38848+86.38848=$475.58. So a preferred embodiment of amulti-state allocation in this example has effected an allocation amongthe constituent states so that (1) the desired payout distributions inthis example are achieved, i.e., payouts to constituent states are thesame no matter which constituent state occurs, and (2) furtherreallocation iterations of multi-state investments do not change therelative amounts invested across the distribution of states for all themulti-state trades.

Example 3.1.3: Alternate Price Distributions

Assumptions regarding the likely distribution of traded amounts for agroup of DBAR contingent claims may be used, for example, to computereturns for each defined state per unit of amount invested at thebeginning of a trading period (“opening returns”). For various reasons,the amount actually invested in each defined state may not reflect theassumptions used to calculate the opening returns. For instance,investors may speculate that the empirical distribution of returns overthe time horizon may differ from the no-arbitrage assumptions typicallyused in option pricing. Instead of a lognormal distribution, moreinvestors might make investments expecting returns to be significantlypositive rather than negative (perhaps expecting favorable news). InExample 3.1.1, for instance, if traders invested more in states above$85 for the price of MSFT common stock, the returns to states below $85could therefore be significantly higher than returns to states above$85.

In addition, it is well known to derivatives traders that traded optionprices indicate that price distributions differ markedly fromtheoretical lognormality or similar theoretical distributions. Theso-called volatility skew or “smile” refers to out-of-the-money put andcall options trading at higher implied volatilities than options closerto the money. This indicates that traders often expect the distributionof prices to have greater frequency or mass at the extreme observationsthan predicted according to lognormal distributions. Frequently, thiseffect is not symmetric so that, for example, the probability of largelower price outcomes are higher than for extreme upward outcomes.Consequently, in a group of DBAR contingent claims of the presentinvention, investment in states in these regions may be more prevalentand, therefore, finalized returns on outcomes in those regions lower.For example, using the basic DBAR contingent information from Example3.1.1 the following returns may prevail due to investor expectations ofreturn distributions that have more frequent occurrences than thosepredicted by a lognormal distribution, and thus are skewed to the lowerpossible returns. In statistical parlance, such a distribution exhibitshigher kurtosis and negative skewness in returns than the illustrativedistribution used in Example 3.1.1 and reflected in Table 3.1.1-1.

TABLE 3.1.3-1 DBAR Contingent Claim Returns Illustrating NegativelySkewed and Leptokurtotic Return Distribution Amount Invested in ReturnPer Unit States State (′000) if State Occurs (0, 80] 3,150 30.746 (80,80.5] 1,500 65.667 (80.5, 81] 1,600 61.5 (81, 81.5] 1,750 56.143 (81.5,82] 2,100 46.619 (82, 82.5] 2,550 38.216 (82.5, 83] 3,150 30.746 (83,83.5] 3,250 29.769 (83.5, 84] 3,050 31.787 (84, 84.5] 8,800 10.363(84.5, 85] 14,300 5.993 (85, 85.5] 10,950 8.132 (85.5, 86] 11,300 7.85(86, 86.5] 10,150 8.852 (86.5, 87] 11,400 7.772 (87, 87.5] 4,550 20.978(87.5, 88] 1,350 73.074 (88, 88.5] 1,250 79.0 (88.5, 89] 1,150 85.957(89, 89.5] 700 141.857 (89.5, 90] 650 152.846 (90, ∞] 1,350 73.074

The type of complex distribution illustrated in Table 3.1.3-1 isprevalent in the traditional markets. Derivatives traders, actuaries,risk managers and other traditional market participants typically usesophisticated mathematical and analytical tools in order to estimate thestatistical nature of future distributions of risky market outcomes.These tools often rely on data sets (e.g., historical time series,options data) that may be incomplete or unreliable. An advantage of thesystems and methods of the present invention is that such analyses fromhistorical data need not be complicated, and the full outcomedistribution for a group of DBAR contingent claims based on any givenevent is readily available to all traders and other interested partiesnearly instantaneously after each investment.

Example 3.1.4: States Defined For Return Uniformity

It is also possible in preferred embodiments of the present invention todefine states for a group of DBAR contingent claims with irregular orunevenly distributed intervals, for example, to make the traded amountacross the states more liquid or uniform. States can be constructed froma likely estimate of the final distribution of invested amounts in orderto make the likely invested amounts, and hence the returns for eachstate, as uniform as possible across the distribution of states. Thefollowing table illustrates the freedom, using the event and tradingperiod from Example 3.1.1, to define states so as to promoteequalization of the amount likely to be invested in each state.

TABLE 3.1.4-1 State Definition to Make Likely Demand Uniform AcrossStates Invested Amount in Return Per Unit States State (′000) if StateOccurs (0, 81.403] 5,000 19 (81.403, 82.181] 5,000 19 (82.181, 82.71]5,000 19 (82.71, 83.132] 5,000 19 (83.132, 83.497] 5,000 19 (83.497,83.826] 5,000 19 (83.826, 84.131] 5,000 19 (84.131, 84.422] 5,000 19(84.422, 84.705] 5,000 19 (84.705, 84.984] 5,000 19 (84.984, 85.264]5,000 19 (85.264, 85.549] 5,000 19 (85.549, 85.845] 5,000 19 (85.845,86.158] 5,000 19 (86.158, 86.497] 5,000 19 (86.497, 86.877] 5,000 19(86.877, 87.321] 5,000 19 (87.321, 87.883] 5,000 19 (87.883, 88.722]5,000 19 (88.722, ∞] 5,000 19

If investor expectations coincide with the often-used assumption of thelognormal distribution, as reflected in this example, then investmentactivity in the group of contingent claims reflected in Table 3.1.4-1will converge to investment of the same amount in each of the 20 statesidentified in the table. Of course, actual trading will likely yieldfinal market returns which deviate from those initially chosen forconvenience using a lognormal distribution.

Example 3.1.5: Government Bond—Uniformly Constructed States

The event, defined states predetermined termination criteria and otherrelevant data for an illustrative group of DBAR contingent claims basedon a U.S. Treasury Note are set forth below:

Underlying Security: United States Treasury Note, 5.5%, May 31, 2003Bond Settlement Date: Jun. 25, 1999 Bond Maturity Date: May 31, 2003Contingent Claim Expiration: Jul. 2, 1999, Market Close, 4:00 p.m. ESTTrading Period Start Date: Jun. 25, 1999, 4:00 p.m., EST Trading PeriodEnd Date: Jun. 28, 1999, 4:00 p.m., EST Next Trading Period Open: Jun.28, 1999, 4:00 p.m., EST Next Trading Period Close Jun. 29, 1999, 4:00p.m., EST Event: Closing Composite Price as reported on Bloomberg atClaim Expiration Trading Time: 1 day Duration from TED: 5 days Coupon:5.5% Payment Frequency: Semiannual Daycount Basis: Actual/ActualDividends Payable over Time Horizon: 2.75 per 100 on Jun. 30, 1999Treasury note repo rate over Time 4.0% (Actual/360 daycount) Horizon:Spot Price: 99.8125 Forward Price at Expiration: 99.7857 PriceVolatility: 4.7% Trade and Payout Units: U.S. Dollars Total Demand inCurrent Trading $50 million Period: Transaction Fee: 25 basis points(.0025%)

TABLE 3.1.5-1 DBAR Contingent Claims on U.S. Government Note StatesInvestment in State ($) Unit Return if State Occurs (0, 98] 139690.1635356.04 (98, 98.25] 293571.7323 168.89 (98.25, 98.5] 733769.9011 66.97(98.5, 98.75] 1574439.456 30.68 (98.75, 99] 2903405.925 16.18 (99, 99.1]1627613.865 29.64 (99.1, 99.2] 1914626.631 25.05 (99.2, 99.3]2198593.057 21.68 (99.3, 99.4] 2464704.885 19.24 (99.4, 99.5]2697585.072 17.49 (99.5, 99.6] 2882744.385 16.30 (99.6, 99.7]3008078.286 15.58 (99.7, 99.8] 3065194.576 15.27 (99.8, 99.9]3050276.034 15.35 (99.9, 100] 2964602.039 15.82 (100, 100.1] 2814300.65716.72 (100.1, 100.2] 2609637.195 18.11 (100.2, 100.3] 2363883.036 20.10(100.3, 100.4] 2091890.519 22.84 (100.4, 100.5] 1808629.526 26.58(100.5, 100.75] 3326547.254 13.99 (100.75, 101] 1899755.409 25.25 (101,101.25] 941506.1374 51.97 (101.25, 101.5] 405331.6207 122.05 (101.5, ∞]219622.6373 226.09

This Example 3.1.5 and Table 3.1.5-1 illustrate how readily the methodsand systems of the present invention may be adapted to sources of risk,whether from stocks, bonds, or insurance claims. Table 3.1.5-1 alsoillustrates a distribution of defined states which is irregularlyspaced—in this case finer toward the center of the distribution andcoarser at the ends—in order to increase the amount invested in theextreme states.

Example 3.1.6: Outperformance Asset Allocation—Uniform Range

One of the advantages of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimsbased on multiple events and their inter-relationships. For example,many index fund money managers often have a fundamental view as towhether indices of high quality fixed income securities will outperformmajor equity indices. Such opinions normally are contained within amanager's model for allocating funds under management between the majorasset classes such as fixed income securities, equities, and cash.

This Example 3.1.6 illustrates the use of a preferred embodiment of thesystems and methods of the present invention to hedge the real-worldevent that one asset class will outperform another. The illustrativedistribution of investments and calculated opening returns for the groupof contingent claims used in this example are based on the assumptionthat the levels of the relevant asset-class indices are jointlylognormally distributed with an assumed correlation. By defining a groupof DBAR contingent claims on a joint outcome of two underlying events,traders are able to express their views on the co-movements of theunderlying events as captured by the statistical correlation between theevents. In this example, the assumption of a joint lognormaldistribution means that the two underlying events are distributed asfollows:

$\begin{matrix}{{\overset{\sim}{V}}_{\theta}^{1} = {\left( {\frac{V_{\tau}^{1}}{Z^{1}\left( {\tau,\theta} \right)} - \frac{D^{1}\left( {\tau,\theta} \right)}{Z^{1}\left( {\tau,\theta} \right)}} \right)*{\mathbb{e}}^{{{- \sigma_{1}^{2}}/2}*{({\theta - \tau})}}*{\mathbb{e}}^{\sigma_{1}*\sqrt{\theta - \tau}*{dz}_{1}}}} \\{{\overset{\sim}{V}}_{\theta}^{2} = {\left( {\frac{V_{\tau}^{2}}{Z^{2}\left( {\tau,\theta} \right)} - \frac{D^{2}\left( {\tau,\theta} \right)}{Z^{2}\left( {\tau,\theta} \right)}} \right)*{\mathbb{e}}^{{{- \sigma_{2}^{2}}/2}*{({\theta - \tau})}}*{\mathbb{e}}^{\sigma_{2}*\sqrt{\theta - \tau}*{dz}_{2}}}} \\{{g\left( {{dz}_{1},{dz}_{2}} \right)} = {\frac{1}{2*\pi*\sqrt{1 - \rho^{2}}}*{\exp\left( {- \frac{\left( {{dz}_{1}^{2} + {dz}_{2}^{2} - {2*\rho*{dz}_{1}*{dz}_{1}}} \right)}{2*\left( {1 - \rho^{2}} \right)}} \right)}}}\end{matrix}$where the subscripts and superscripts indicate each of the two events,and g(dz₁,dz₂) is the bivariate normal distribution with correlationparameter ρ, and the notation otherwise corresponds to the notation usedin the description above of DBAR Range Derivatives.

The following information includes the indices, the trading periods, thepredetermined termination criteria, the total amount invested and thevalue units used in this Example 3.1.6:

Asset Class 1: JP Morgan United States Government Bond Index (“JPMGBI”)Asset Class 1 Forward Price at Observation: 250.0 Asset Class 1Volatility: 5% Asset Class 2: S&P 500 Equity Index (“SP500”) Asset Class2 Forward Price at Observation: 1410 Asset Class 2 Volatility: 18%Correlation Between Asset Classes: 0.5 Contingent Claim Expiration: Dec.31, 1999 Trading Start Date: Jun. 30, 1999 Current Trading Period StartDate: Jul. 1, 1999 Current Trading Period End Date: Jul. 30, 1999 NextTrading Period Start Date: Aug. 2, 1999 Next Trading Period End Date:Aug. 31, 1999 Current Date: Jul. 12, 1999 Last Trading Period End Date:Dec. 30, 1999 Aggregate Investment for Current Trading $100 millionPeriod: Trade and Payout Value Units: U.S. DollarsTable 3.1.6 shows the illustrative distribution of state returns overthe defined states for the joint outcomes based on this information,with the defined states as indicated.

TABLE 3.1.6-1 Unit Returns for Joint Performance of S & P 500 and JPMGBIJPMGBI (233, (237, (241, (244, (246, (248, (250, (252, (255, (257, (259,(264, (268, State (0, 233] 237] 241] 244] 246] 248] 250] 252] 255] 257]259] 264] 268] ∞] (0, 1102] 246 240 197 413 475 591 798 1167 1788 30393520 2330 11764 18518 (1102, 1174] 240 167 110 197 205 230 281 373 538841 1428 1753 7999 11764 (1174, 1252] 197 110 61 99 94 98 110 135 180259 407 448 1753 5207 (1252, 1292] 413 197 99 145 130 128 136 157 197269 398 407 1428 5813 (1292, 1334] 475 205 94 130 113 106 108 120 144189 269 259 841 3184 (1334, 1377] 591 230 98 128 106 95 93 99 115 144197 180 538 1851 SP500 (1377, 1421] 798 281 110 136 108 93 88 89 99 120157 135 373 1167 (1421, 1467] 1167 373 135 157 120 99 89 88 93 108 136110 281 798 (1467, 1515] 1851 538 180 197 144 115 99 93 95 106 128 98230 591 (1515, 1564] 3184 841 259 269 189 144 120 108 106 113 130 94 205475 (1564, 1614] 5813 1428 407 398 269 197 157 136 128 130 145 99 197413 (1614, 1720] 5207 1753 448 407 259 180 135 110 98 94 99 61 110 197(1720, 1834] 11764 7999 1753 1428 841 538 373 281 230 205 197 110 167240 (1834, ∞] 18518 11764 2330 3520 3039 1788 1167 798 591 475 413 197240 246

In Table 3.1.6-1, each cell contains the unit returns to the joint statereflected by the row and column entries. For example, the unit return toinvestments in the state encompassing the joint occurrence of the JPMGBIclosing on expiration at 249 and the SP500 closing at 1380 is 88. Sincethe correlation between two indices in this example is assumed to be0.5, the probability both indices will change in the same direction isgreater that the probability that both indices will change in oppositedirections. In other words, as represented in Table 3.1.6-1, unitreturns to investments in states represented in cells in the upper leftand lower right of the table—i.e., where the indices are changing in thesame direction—are lower, reflecting higher implied probabilities, thanunit returns to investments to states represented in cells in the lowerleft and upper right of Table 3.1.6-1—i.e., where the indices arechanging in opposite directions.

As in the previous examples and in preferred embodiments, the returnsillustrated in Table 3.1.6-1 could be calculated as opening indicativereturns at the start of each trading period based on an estimate of whatthe closing returns for the trading period are likely to be. Theseindicative or opening returns can serve as an “anchor point” forcommencement of trading in a group of DBAR contingent claims. Of course,actual trading and trader expectations may induce substantial departuresfrom these indicative values.

Example 3.1.7: Corporate Bond Credit Risk

Groups of DBAR contingent claims can also be constructed on creditevents, such as the event that one of the major credit rating agencies(e.g., Standard and Poor's, Moodys) changes the rating for some or allof a corporation's outstanding securities. Indicative returns at theoutset of trading for a group of DBAR contingent claims oriented to acredit event can readily be constructed from publicly available datafrom the rating agencies themselves. For example, Table 3.1.7-1 containsindicative returns for an assumed group of DBAR contingent claims basedon the event that a corporation's Standard and Poor's credit rating fora given security will change over a certain period of time. In thisexample, states are defined using the Standard and Poor's creditcategories, ranging from AAA to D (default). Using the methods of thepresent invention, the indicative returns are calculated usinghistorical data on the frequency of the occurrence of these definedstates. In this example, a transaction fee of 1% is charged against theaggregate amount invested in the group of DBAR contingent claims, whichis assumed to be $100 million.

TABLE 3.1.7-1 Illustrative Returns for Credit DBAR Contingent Claimswith 1% Transaction Fee Current To New Historical Invested in IndicativeReturn to Rating Rating Probability State ($) State A− AAA 0.0016160,000 617.75 A− AA+ 0.0004 40,000 2474.00 A− AA 0.0012 120,000 824.00A− AA− 0.003099 309,900 318.46 A− A+ 0.010897 1,089,700 89.85 A− A0.087574 8,757,400 10.30 A− A− 0.772868 77,286,800 0.28 A− BBB+ 0.0689796,897,900 13.35 A− BBB 0.03199 3,199,000 29.95 A− BBB- 0.007398 739,800132.82 A− BB+ 0.002299 229,900 429.62 A− BB 0.004999 499,900 197.04 A−BB- 0.002299 229,900 429.62 A− B+ 0.002699 269,900 365.80 A− B 0.000440,000 2474.00 A− B− 0.0004 40,000 2474.00 A− CCC 1E−04 10,000 9899.00A− D 0.0008 80,000 1236.50

In Table 3.1.7-1, the historical probabilities over the mutuallyexclusive and collectively exhaustive states sum to unity. Asdemonstrated above in this specification, in preferred embodiments, thetransaction fee affects the probability implied for each state from theunit return for that state.

Actual trading is expected almost always to alter illustrativeindicative returns based on historical empirical data. This Example3.1.7 indicates how efficiently groups of DBAR contingent claims can beconstructed for all traders or firms exposed to particular credit riskin order to hedge that risk. For example, in this Example, if a traderhas significant exposure to the A-rated bond issue described above, thetrader could want to hedge the event corresponding to a downgrade byStandard and Poor's. For example, this trader may be particularlyconcerned about a downgrade corresponding to an issuer default or “D”rating. The empirical probabilities suggest a payout of approximately$1,237 for each dollar invested in that state. If this trader has$100,000,000 of the corporate issue in his portfolio and a recovery ofratio of 0.3 can be expected in the event of default, then, in order tohedge $70,000,000 of default risk, the trader might invest in the stateencompassing a “D” outcome. To hedge the entire amount of the defaultrisk in this example, the amount of the investment in this state shouldbe $70,000,000/$1,237 or $56,589. This represents approximately 5.66basis points of the trader's position size in this bond (i.e.,$56,589/$100,000,000=0.00056)] which probably represents a reasonablecost of credit insurance against default. Actual investments in thisgroup of DBAR contingent claims could alter the return on the “D” eventover time and additional insurance might need to be purchased.

Example 3.1.8: Economic Statistics

Another advantage of the methods and systems of the present invention isthat they allow market participants to hedge possible outcomes overevents which cannot be hedged directly in traditional derivativesmarkets. For example, traders often hedge inflation risk by trading inbond futures or, where they exist, inflation-protected floating ratebonds. A group of DBAR contingent claims can readily be constructed toallow traders to express expectations about the distribution ofuncertain economic statistics measuring, for example, the rate ofinflation or other relevant variables. The following informationdescribes such a group of claims:

Economic Statistic: United States Non-Farm Payrolls Announcement Date:May 31, 1999 Last Announcement Date: Apr. 30, 1999 Expiration:Announcement Date, May 31, 1999 Trading Start Date: May 1, 1999 CurrentTrading Period Start Date: May 10, 1999 Current Trading Period End Date:May 14, 1999 Current Date: May 11, 1999 Last Announcement: 128,156(′000) Source: Bureau of Labor Statistics Consensus Estimate: 130,000(+1.2%) Aggregate Amount Invested in $100 million Current Period:Transaction Fee: 2.0% of Aggregate Traded amount

Using methods and systems of the present invention, states can bedefined and indicative returns can be constructed from, for example,consensus estimates among economists for this index. These estimates canbe expressed in absolute values or, as illustrated, in Table 3.1.8-1 inpercentage changes from the last observation as follows:

TABLE 3.1.8-1 Illustrative Returns For Non-Farm Payrolls Release with 2%Transaction Fee % Chg. In Index Investment in State Implied State State(′000) State Returns Probability (−100, −5] 100 979 0.001 (−5, −3] 200489 0.002 (−3, −1] 400 244 0.004 (−1, −.5] 500 195 0.005 (−.5, 0] 100097 0.01 (0, .5] 2000 48 0.02 (.5, .7] 3000 31.66667 0.03 (.7, .8] 400023.5 0.04 (.8, .9] 5000 18.6 0.05 (.9, 1.0] 10000 8.8 0.1 (1.0, 1.1]14000 6 0.14 (1.1, 1.2] 22000 3.454545 0.22 (1.2, 1.25] 18000 4.4444440.18 (1.25, 1.3] 9000 9.888889 0.09 (1.3, 1.35] 6000 15.33333 0.06(1.35, 1.40] 3000 31 .66667 0.03 (1.40, 1.45] 200 489 0.002 (1.45, 1.5]600 162.3333 0.006 (1.5, 1.6] 400 244 0.004 (1.6, 1.7] 100 979 0.001(1.7, 1.8] 80 1224 0.0008 (1.8, 1.9] 59 1660.017 0.00059 (1.9, 2.0] 591660.017 0.00059 (2.0, 2.1] 59 1660.017 0.00059 (2.1, 2.2] 59 1660.0170.00059 (2.2, 2.4] 59 1660.017 0.00059 (2.4, 2.6] 59 1660.017 0.00059(2.6, 3.0] 59 1660.017 0.00059 (3.0, ∞] 7 13999 0.00007As in examples, actual trading prior to the trading end date would beexpected to adjust returns according to the amounts invested in eachstate and the total amount invested for all the states.

Example 3.1.9: Corporate Events Corporate actions and announcements arefurther examples of events of economic significance which are usuallyunhedgable or uninsurable in traditional markets but which can beeffectively structured into groups of DBAR contingent claims accordingto the present invention. Examples of such corporate events are earningsannouncements, which typically occur quarterly for publicly tradedcompanies. Many data services, such as IBES and FirstCall, currentlypublish estimates by analysts and a consensus estimate in advance ofquarterly earnings announcements. Such estimates can form the basis forindicative opening returns at the commencement of trading as illustratedbelow. For this example, a transaction fee of zero is assumed.

Underlying security: IBM Earnings Announcement Date: Jul. 21, 1999Consensus Estimate: .879/share Expiration: Announcement, Jul. 21, 1999First Trading Period Start Date: Apr. 19, 1999 First Trading Period EndDate May 19, 1999 Current Trading Period Start Date: Jul. 6, 1999Current Trading Period End Date: Jul. 9, 1999 Next Trading Period StartDate: Jul. 9, 1999 Next Trading Period End Date: Jul. 16, 1999 TotalAmount Invested in Current Trading Period: $100 million

TABLE 3.1.9-1 Illustrative Returns For IBM Earnings AnnouncementEarnings Invested in State State 0 (′000 $) Unit Returns Implied StateProbability (−∞, .5] 70 1,427.57 0.0007 (.5, .6] 360 276.78 0.0036 (.6,.65] 730 135.99 0.0073 (.65, .7] 1450 67.97 0.0145 (.7, .74] 2180 44.870.0218 (.74, .78] 3630 26.55 0.0363 (.78, .8] 4360 21.94 0.0436 (.8,.82] 5820 16.18 0.0582 (.82, .84] 7270 12.76 0.0727 (.84, .86] 872010.47 0.0872 (.86, .87] 10900 8.17 0.109 (.87, .88] 18170 4.50 0.1817(.88, .89] 8720 10.47 0.0872 (.89, .9] 7270 12.76 0.0727 (.9, .91] 509018.65 0.0509 (.91, .92] 3630 26.55 0.0363 (.92, .93] 2910 33.36 0.0291(.93, .95] 2180 44.87 0.0218 (.95, .97] 1450 67.97 0.0145 (.97, .99]1310 75.34 0.0131 (.99, 1.1] 1160 85.21 0.0116 (1.1, 1.3] 1020 97.040.0102 (1.3, 1.5] 730 135.99 0.0073 (1.5, 1.7] 360 276.78 0.0036 (1.7,1.9] 220 453.55 0.0022 (1.9, 2.1] 150 665.67 0.0015 (2.1, 2.3] 701,427.57 0.0007 (2.3, 2.5] 40 2,499.00 0.0004 (2.5, ∞] 30 3332.33 0.0003Consistent with the consensus estimate, the state with the largestinvestment encompasses the range (0.87, 0.88].

Example 3.1.10: Real Assets

Another advantage of the methods and systems of the present invention isthe ability to structure liquid claims on illiquid underlying assetssuch a real estate. As previously discussed, traditional derivativesmarkets customarily use a liquid underlying market in order to functionproperly. With a group of DBAR contingent claims all that is usuallyrequired is a real-world, observable event of economic significance. Forexample, the creation of contingent claims tied to real assets has beenattempted at some financial institutions over the last several years.These efforts have not been credited with an appreciable impact,apparently because of the primary liquidity constraints inherent in theunderlying real assets.

A group of DBAR contingent claims according to the present invention canbe constructed based on an observable event related to real estate. Therelevant information for an illustrative group of such claims is asfollows:

Real Asset Index: Colliers ABR Manhattan Office Rent Rates BloombergTicker: COLAMANR Update Frequency: Monthly Source: Colliers ABR, Inc.Announcement Date: Jul. 31, 1999 Last Announcement Date: Jun. 30, 1999Last Index Value: $45.39/sq. ft. Consensus Estimate: $45.50 Expiration:Announcement Jul. 31, 1999 Current Trading Period Start: Jun. 30, 1999Current Trading Period End: Jul. 7, 1999 Next Trading Period Start Jul.7, 1999 Next Trading Period End Jul. 14, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be calculated or willemerge from actual trader investments according to the methods of thepresent invention as illustrated in Examples 3.1.1-3.1.9.

Example 3.1.11: Energy Supply Chain

A group of DBAR contingent claims can also be constructed using themethods and systems of the present invention to provide hedging vehicleson non-tradable quantities of great economic significance within thesupply chain of a given industry. An example of such an application isthe number of oil rigs currently deployed in domestic U.S. oilproduction. The rig count tends to be a slowly adjusting quantity whichis sensitive to energy prices. Thus, appropriately structured groups ofDBAR contingent claims based on rig counts could enable suppliers,producers and drillers to hedge exposure to sudden changes in energyprices and could provide a valuable risk-sharing device.

For example, a group of DBAR contingent claims depending on the rigcount could be constructed according to the present invention using thefollowing information (e.g., data source, termination criteria, etc).

Asset Index: Baker Hughes Rig Count U.S. Total Bloomberg Ticker: BAKETOTFrequency: Weekly Source: Baker Hughes, Inc. Announcement Date: Jul. 16,1999 Last Announcement Date: Jul. 9, 1999 Expiration Date: Jul. 16, 1999Trading Start Date: Jul. 9, 1999 Trading End Date: Jul. 15, 1999 Last:570 Consensus Estimate: 580

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

Example 3.1.12: Mortgage Prepayment Risk

Real estate mortgages comprise an extremely large fixed income assetclass with hundreds of billions in market capitalization. The mortgagemarket is generally understood to be subject not only to interest raterisk but also to the risk that borrowers will exercise options torefmance their mortgages or otherwise “prepay” their existing mortgageloans. The owner of a mortgage security therefore bears the risk that hewill be “called” out of his position when the mortgage interest ratelevels are declining. This risk cannot readily be hedged directly inexisting markets. This risk could, however be hedged or insured withgroups of DBAR contingent claims structured according to the presentinvention. Such a group of contingent claims could, for example, bestructured based on the following information:

Asset Index: FNMA Conventional 30 year One-Month Historical AggregatePrepayments Coupon: 6.5% Frequency: Monthly Source: BloombergAnnouncement Date: Aug. 1, 1999 Last Announcement Date: Jul. 1, 1999Expiration: Announcement Date, Aug. 1, 1999 Current Trading Period StartJul. 1, 1999 Date: Current Trading Period End Jul. 9, 1999 Date: Last:303 Public Securities Association Prepayment Speed (“PSA”) ConsensusEstimate: 310 PSA

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

Example 3.1.13: Insurance Industry Loss Warranty (“ILW”)

Groups of DBAR contingent claims can also be structured using the systemand methods of the present invention to provide insurance andreinsurance facilities for property and casualty, life, health and othertraditional lines of insurance. The following information providesinformation to structure a group of DBAR contingent claims related tolarge property losses from hurricane damage:

Event: PCS Eastern Excess $5 billion Index Source: Property ClaimServices (PCS) Frequency: Monthly Announcement Date: Oct. 1, 1999 LastAnnouncement Date: Jul. 1, 1999 Last Index Value: No events ConsensusEstimate: $1 billion (claims excess of $5 billion) Expiration:Announcement Date, Oct. 1, 1999 Trading Period Start Date: Jul. 1, 1999Trading Period End Date: Sep. 30, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

In preferred embodiments of groups of DBAR contingent claims related toproperty-casualty catastrophe losses, the frequency of claims and thedistributions of the severity of losses are assumed and convolutions areperformed in order to post indicative returns over the distribution ofdefined states. This can be done, for example, using compoundfrequency-severity models, such as the Poisson-Pareto model, familiar tothose of skill in the art, which predict, with greater probability thana normal distribution, that losses will be extreme. As indicatedpreviously, in preferred embodiments market activity is expected toalter the posted indicative returns, which serve as informative levelsat the commencement of trading.

Example 3.1.14: Conditional Events

As discussed above, advantage of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimsrelated to events of economic significance for which there is greatinterest in insurance and hedging, but which are not readily hedged orinsured in traditional capital and insurance markets. Another example ofsuch an event is one that occurs only when some related event haspreviously occurred. For purposes of illustration, these two events maybe denoted A and B.

${q\left\langle A \middle| B \right\rangle} = \frac{q\left( {A\bigcap B} \right)}{q(B)}$where q denotes the probability of a state, q<A|B> represents theconditional probability of state A given the prior occurrence of stateand B, and q(A∩B) represents the occurrence of both states A and B.

For example, a group of DBAR contingent claims may be constructed tocombine elements of “key person” insurance and the performance of thestock price of the company managed by the key person. Many firms aremanaged by people whom capital markets perceive as indispensable orparticularly important, such as Warren Buffett of Berkshire Hathaway.The holders of Berkshire Hathaway stock have no ready way of insuringagainst the sudden change in management of Berkshire, either due to acorporate action such as a takeover or to the death or disability ofWarren Buffett. A group of conditional DBAR contingent claims can beconstructed according to the present invention where the defined statesreflect the stock price of Berkshire Hathaway conditional on WarrenBuffet's leaving the firm's management. Other conditional DBARcontingent claims that could attract significant amounts for investmentcan be constructed using the methods and systems of the presentinvention, as apparent to one of skill in the art.

Example 3.1.15: Securitization Using a DBAR Contingent Claim Mechanism

The systems and methods of the present invention can also be adapted bya financial intermediary or issuer for the issuance of securities suchas bonds, common or preferred stock, or other types of financialinstruments. The process of creating new opportunities for hedgingunderlying events through the creation of new securities is known as“securitization.” Well-known examples of securitization include themortgage and asset-backed securities markets, in which portfolios offinancial risk are aggregated and then recombined into new sources offinancial risk. The systems and methods of the present invention can beused within the securitization process by creating securities, orportfolios of securities, whose risk, in whole or part, is tied to anassociated or embedded group of DBAR contingent claims. In a preferredembodiment, a group of DBAR contingent claims is associated with asecurity much like options are currently associated with bonds in orderto create callable and putable bonds in the traditional markets.

This example illustrates how a group of DBAR contingent claims accordingto the present invention can be tied to the issuance of a security inorder to share risk associated with an identified future event among thesecurity holders. In this example, the security is a fixed income bondwith an embedded group of DBAR contingent claims whose value depends onthe possible values for hurricane losses over some time period for somegeographic region.

Issuer: Tokyo Fire and Marine Underwriter: Goldman Sachs DBAR Event:Total Losses on a Saffir-Simpson Category 4 Hurricane Geographic:Property Claims Services Eastern North America Date: Jul. 1, 1999-Nov.1, 1999 Size of Issue: 500 million USD. Issue Date: Jun. 1, 1999 DBARTrading Period: Jun. 1, 1999-Jul. 1, 1999

In this example, the underwriter Goldman Sachs issues the bond, andholders of the issued bond put bond principal at risk over the entiredistribution of amounts of Category 4 losses for the event. Ranges ofpossible losses comprise the defined states for the embedded group ofDBAR contingent claims. In a preferred embodiment, the underwriter isresponsible for updating the returns to investments in the variousstates, monitoring credit risk, and clearing and settling, andvalidating the amount of the losses.

When the event is determined and uncertainty is resolved, Goldman is“put” or collects the bond principal at risk from the unsuccessfulinvestments and allocates these amounts to the successful investments.The mechanism in this illustration thus includes:

-   -   (1) An underwriter or intermediary which implements the        mechanism, and    -   (2) A group of DBAR contingent claims directly tied to a        security or issue (such as the catastrophe bond above).

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in Examples 3.1.1-3.1.9.

Example 3.1.16: Exotic Derivatives

The securities and derivatives communities frequently use the term“exotic derivatives” to refer to derivatives whose values are linked toa security, asset, financial product or source of financial risk in amore complicated fashion than traditional derivatives such as futures,call options, and convertible bonds. Examples of exotic derivativesinclude American options, Asian options, barrier options, Bermudanoptions, chooser and compound options, binary or digital options,lookback options, automatic and flexible caps and floors, and shoutoptions.

Many types of exotic options are currently traded. For example, barrieroptions are rights to purchase an underlying financial product, such asa quantity of foreign currency, for a specified rate or price, but onlyif, for example, the underlying exchange rate crosses or does not crossone or more defined rates or “barriers.” For example, a dollar call/yenput on the dollar/yen exchange rate, expiring in three months withstrike price 110 and “knock-out” barrier of 105, entitles the holder topurchase a quantity of dollars at 110 yen per dollar, but only if theexchange rate did not fall below 105 at any point during the three monthduration of the option. Another example of a commonly traded exoticderivative, an Asian option, depends on the average value of theunderlying security over some time period. Thus, a class of exoticderivatives is commonly referred to as “path-dependent” derivatives,such as barrier and Asian options, since their values depend not only onthe value of the underlying financial product at a given date, but on ahistory of the value or state of the underlying financial product.

The properties and features of exotic derivatives are often so complexso as to present a significant source of “model risk” or the risk thatthe tools, or the assumptions upon which they are based, will lead tosignificant errors in pricing and hedging. Accordingly, derivativestraders and risk managers often employ sophisticated analytical tools totrade, hedge, and manage the risk of exotic derivatives.

One of the advantages of the systems and methods of the presentinvention is the ability to construct groups of DBAR contingent claimswith exotic features which are more manageable and transparent thantraditional exotic derivatives. For example, a trader might be solelyinterested in the earliest time the yen/dollar exchange rate crosses 95over the next three months. A traditional barrier option, or portfolioof such exotic options, might suffice to approximate the source of riskof interest to this trader. A group of DBAR contingent claims, incontrast, can be constructed to isolate this risk and present relativelytransparent opportunities for hedging. A risk to be isolated is thedistribution of possible outcomes for what barrier derivatives tradersterm the “first passage time,” or, in this example, the first time thatthe yen/dollar exchange rate crosses 95 over the next three months.

The following illustration shows how such a group of DBAR contingentclaims can be constructed to address this risk. In this example, it isassumed that all traders in the group of claims agree that theunderlying exchange rate is lognormally distributed. This group ofclaims illustrates how traders would invest in states and thus expressopinions regarding whether and when the forward yen/dollar exchange ratewill cross a given barrier over the next 3 months:

Underlying Risk: Japanese/U.S. Dollar Yen Exchange Rate Current Date:Sep. 15, 1999 Expiration: Forward Rate First Passage Time, as defined,between Sep. 16, 1999 to Dec. 16, 1999 Trading Start Date: Sep. 15, 1999Trading End Date: Sep. 16, 1999 Barrier: 95 Spot JPY/USD: 104.68 ForwardJPY/USD 103.268 Assumed (Illustrative) 20% annualized Market Volatility:Aggregate Traded Amount: 10 million USD

TABLE 3.1.16-1 First Passage Time for Yen/Dollar Dec. 16, 1999 ForwardExchange Rate Return Per Unit if Time in Year Fractions Invested inState (′000) State Occurs (0, .005] 229.7379 42.52786 (.005, .01]848.9024 10.77992 (.01, .015] 813.8007 11.28802 (.015, .02] 663.216514.07803 (.02, .025] 536.3282 17.6453 (.025, .03] 440.5172 21.70059(.03, .035] 368.4647 26.13964 (.035, .04] 313.3813 30.91 (.04, .045]270.4207 35.97942 (.045, .05] 236.2651 41 .32534 (.05, .075] 850.259510.76112 (.075, .1] 540.0654 17.51627 (.1, .125] 381.3604 25.22191(.125, .15] 287.6032 33.77013 (.15, .175] 226.8385 43.08423 (.175, .2]184.8238 53.10558 (.2, .225] 154.3511 63.78734 (.225, .25] 131 .421775.09094 Did Not Hit Barrier 2522.242 2.964727

As with other examples, and in preferred embodiments, actual tradingwill likely generate traded amounts and therefore returns that departfrom the assumptions used to compute the illustrative returns for eachstate.

Example 3.1.17: Hedging Markets for Real Goods, Commodities and Services

Investment and capital budgeting choices faced by firms typicallyinvolve inherent economic risk (e.g., future demand for semiconductors),large capital investments (e.g., semiconductor fabrication capacity) andtiming (e.g., a decision to invest in a plant now, or defer for someperiod of time). Many economists who study such decisions underuncertainty have recognized that such choices involve what they term“real options.” This characterization indicates that the choice toinvest now or to defer an investment in goods or services or a plant,for example, in the face of changing uncertainty and information,frequently entails risks similar to those encountered by traders whohave invested in options which provide the opportunity to buy or sell anunderlying asset in the capital markets. Many economists and investorsrecognize the importance of real options in capital budgeting decisionsand of setting up markets to better manage their uncertainty and value.Natural resource and extractive industries, such as petroleumexploration and production, as well as industries requiring largecapital investments such as technology manufacturing, are prime examplesof industries where real options analysis is increasingly used andvalued.

Groups of DBAR contingent claims according to the present invention canbe used by firms or firms within a given industry to better analyzecapital budgeting decisions, including those involving real options. Forexample, a group of DBAR contingent claims can be established whichprovides hedging opportunities over the distribution of futuresemiconductor prices. Such a group of claims would allow producers ofsemiconductors to better hedge their capital budgeting decisions andprovide information as to the market's expectation of future prices overthe entire distribution of possible price outcomes. This informationabout the market's expectation of future prices could then also be usedin the real options context in order to better evaluate capitalbudgeting decisions. Similarly, computer manufacturers could use suchgroups of DBAR contingent claims to hedge against adverse semiconductorprice changes.

Information providing the basis for constructing an illustrative groupof DBAR contingent claims on semiconductor prices is as follows:

Underlying Event: Semiconductor Monthly Sales Index: SemiconductorIndustry Association Monthly Global Sales Release Current Date: Sep. 15,1999 Last Release Date: Sep. 2, 1999 Last Release Month: July, 1999 LastRelease Value: 11.55 Billion, USD Next Release Date: Approx. Oct. 1,1999 Next Release Month: August 1999 Trading Start Date: Sep. 2, 1999Trading End Date: Sep. 30, 1999

For reasons of brevity, defined states and opening indicative orillustrative returns resulting from amounts invested in the variousstates for this example are not shown, but can be readily calculated orwill emerge from actual trader investments according to the methods ofthe present invention, as illustrated in previous examples.

Groups of DBAR contingent claims according to the present invention canalso be used to hedge arbitrary sources of risk due to price discoveryprocesses. For example, firms involved in competitive bidding for goodsor services, whether by sealed bid or open bid auctions, can hedge theirinvestments and other capital expended in preparing the bid by investingin states of a group of DBAR contingent claims comprising ranges ofmutually exclusive and collectively exhaustive auction bids. In thisway, the group of DBAR contingent claim serves as a kind of“meta-auction,” and allows those who will be participating in theauction to invest in the distribution of possible auction outcomes,rather than simply waiting for the single outcome representing theauction result. Auction participants could thus hedge themselves againstadverse auction developments and outcomes, and, importantly, have accessto the entire probability distribution of bids (at least at one point intime) before submitting a bid into the real auction. Thus, a group ofDBAR claims could be used to provide market data over the entiredistribution of possible bids. Preferred embodiments of the presentinvention thus can help avoid the so-called Winner's Curse phenomenonknown to economists, whereby auction participants fail rationally totake account of the information on the likely bids of their auctioncompetitors.

Example 3.1.18: DBAR Hedging

Another feature of the systems and methods of the present invention isthe relative ease with which traders can hedge risky exposures. In thefollowing example, it is assumed that a group of DBAR contingent claimshas two states (state 1 and state 2, or s₁ or s₂, and amounts T₁, and T₂are invested in state 1 and state 2, respectively. The unit payout π₁for state 1 is therefore T₂/T₁ and for state 2 it is T₁/T₂. If a traderthen invests amount α₁ in state 1, and state 1 then occurs, the traderin this example would receive the following payouts, P, indexed by theappropriate state subscripts:

$P_{1} = {\alpha_{1}*\left( {\frac{T_{2}}{T_{1} + \alpha_{I}} + 1} \right)}$If state 2 occurs the trader would receiveP ₂=0If, at some point during the trading period, the trader desires to hedgehis exposure, the investment in state 2 to do so is calculated asfollows:

$\alpha_{2} = \frac{\alpha_{1}*T_{2}}{T_{1}}$This is found by equating the state payouts with the proposed hedgetrade, as follows:

$P_{1} = {{\alpha_{1}*\left( {\frac{T_{2} + \alpha_{2}}{T_{1} + \alpha_{1}} + 1} \right)} = {P_{2} = {\alpha_{2}*\left( {\frac{T_{1} + \alpha_{1}}{T_{2} + \alpha_{2}} + 1} \right)}}}$

Compared to the calculation required to hedge traditional derivatives,these expressions show that, in appropriate groups of DBAR contingentclaims of the present invention, calculating and implementing hedges canbe relatively straightforward.

The hedge ratio, α₂, just computed for a simple two state example can beadapted to a group of DBAR contingent claims which is defined over morethan two states. In a preferred embodiment of a group of DBAR contingentclaims, the existing investments in states to be hedged can bedistinguished from the states on which a future hedge investment is tobe made. The latter states can be called the “complement” states, sincethey comprise all the states that can occur other than those in whichinvestment by a trader has already been made, i.e., they arecomplementary to the invested states. A multi-state hedge in a preferredembodiment includes two steps: (1) determining the amount of the hedgeinvestment in the complement states and (2) given the amount sodetermined, allocating the amount among the complement states. Theamount of the hedge investment in the complement states pursuant to thefirst step is calculated as:

$\alpha_{C} = \frac{\alpha_{H}*T_{C}}{T_{H}}$where α_(C) is amount of the hedge investment in the complement states,α_(H) is the amount of the existing investment in the states to behedged, T_(C) is the existing amount invested in the complement states,and T_(H) is the amount invested the states to be hedged, exclusive ofα_(H), The second step involves allocating the hedge investment amongthe complement states, which can be done by allocating α_(c), among thecomplement states in proportion to the existing amounts already investedin each of those states.

An example of a four state group of DBAR contingent claims according tothe present invention illustrates this two-step hedging process. Forpurposes of this example, the following assumptions are made: (i) thereare four states, numbered 1 through 4, respectively; (ii) $50, $80, $70and $40 is invested in each state, (iii) a trader has previously placeda multi-state investment in the amount of $10 (α_(H) as defined above)for states 1 and 2; and (iv) the allocation of this multi-stateinvestment in states 1 and 2 is $3.8462 and $6.15385, respectively. Theamounts invested in each state, excluding the trader's invested amounts,are therefore $46.1538, $73.84615, $70, and $40 for states 1 through 4,respectively. It is noted that the amount invested in the states to behedged, i.e., states 1 and 2, exclusive of the multi-state investment of$10, is the quantity T_(H) as defined above.

The first step in a preferred embodiment of the two-step hedging processis to compute the amount of the hedge investment to be made in thecomplement states. As derived above, the amount of the new hedgeinvestment is equal to the amount of the existing investment multipliedby the ratio of the amount invested in the complement states to theamount invested in the states to be hedged excluding the trader'sexisting trades, i.e., $10*($70+$40)/($46.1538+$73.84615)=$9.16667. Thesecond step in this process is to allocate this amount between the twocomplement states, i.e., states 3 and 4.

Following the procedures discussed above for allocating multi-stateinvestments, the complement state allocation is accomplished byallocating the hedge investment amount—$9.16667 in this example—inproportion to the existing amount previously invested in the complementstates, i.e., $9.16667*$70/$110=$5.83333 for state 3 and$9.16667*$40/$110=$3.3333 for state 4. Thus, in this example, the tradernow has the following amounts invested in states 1 through 4: ($3.8462,$6.15385, $5.8333, $3.3333); the total amount invested in each of thefour states is $50, $80, $75.83333, and $43.3333); and the returns foreach of the four states, based on the total amount invested in each ofthe four states, would be, respectively, (3.98333, 2.1146, 2.2857, and4.75). In this example, if state 1 occurs the trader will receive apayout, including the amount invested in state 1, of3.98333*$3.8462+$3.8462=$19.1667 which is equal to the sum invested, sothe trader is fully hedged against the occurrence of state 1.Calculations for the other states yield the same results, so that thetrader in this example would be fully hedged irrespective of which stateoccurs.

As returns can be expected to change throughout the trading period, thetrader would correspondingly need to rebalance both the amount of hishedge investment for the complement states as well as the multi-stateallocation among the complement states. In a preferred embodiment, aDBAR contingent claim exchange can be responsible for reallocatingmulti-state trades via a suspense account, for example, so the tradercan assign the duty of reallocating the multi-state investment to theexchange. Similarly, the trader can also assign to an exchange theresponsibility of determining the amount of the hedge investment in thecomplement states especially as returns change as a result of trading.The calculation and allocation of this amount can be done by theexchange in a similar fashion to the way the exchange reallocatesmulti-state trades to constituent states as investment amounts change.

Example 3.1.19: Quasi-Continuous Trading

Preferred embodiments of the systems and methods of the presentinvention include a trading period during which returns adjust amongdefined states for a group of DBAR contingent claims, and a laterobservation period during which the outcome is ascertained for the eventon which the group of claims is based. In preferred embodiments, returnsare allocated to the occurrence of a state based on the finaldistribution of amounts invested over all the states at the end of thetrading period. Thus, in each embodiments a trader will not know hisreturns to a given state with certainty until the end of a given tradingperiod. The changes in returns or “price discovery” which occur duringthe trading period prior to “locking-in” the final returns may provideuseful information as to trader expectations regarding finalizedoutcomes, even though they are only indications as to what the finalreturns are going to be. Thus, in some preferred embodiments, a tradermay not be able to realize profits or losses during the trading period.The hedging illustration of Example 3.1.18, for instance, provides anexample of risk reduction but not of locking-in or realizing profit andloss.

In other preferred embodiments, a quasi-continuous market for trading ina group of DBAR contingent claims may be created. In preferredembodiments, a plurality of recurring trading periods may providetraders with nearly continuous opportunities to realize profit and loss.In one such embodiment, the end of one trading period is immediatelyfollowed by the opening of a new trading period, and the final investedamount and state returns for a prior trading period are “locked in” asthat period ends, and are allocated accordingly when the outcome of therelevant event is later known. As a new trading period begins on thegroup of DBAR contingent claims related to the same underlying event, anew distribution of invested amounts for states can emerge along with acorresponding new distribution of state returns. In such embodiments, asthe successive trading periods are made to open and close morefrequently, a quasi-continuous market can be obtained, enabling tradersto hedge and realize profit and loss as frequently as they currently doin the traditional markets.

An example illustrates how this feature of the present invention may beimplemented. The example illustrates the hedging of a European digitalcall option on the yen/dollar exchange rate (a traditional marketoption) over a two day period during which the underlying exchange ratechanges by one yen per dollar. In this example, two trading periods areassumed for the group of DBAR contingent claims

Traditional Option: European Digital Option Payout of Option: Pays 100million USD if exchange rate equals or exceeds strike price at maturityUnderlying Index: Yen/dollar exchange rate Option Start: Aug. 12, 1999Option Expiration: Aug. 15, 2000 Assumed Volatility: 20% annualizedStrike Price: 120 Notional: 100 million USD

In this example, two dates are analyzed, Aug. 12, 1999 and Aug. 13,1999:

TABLE 3.1.19-1 Change in Traditional Digital Call Option Value Over TwoDays Observation Date Aug. 12, 1999 Aug. 13, 1999 Spot Settlement DateAug. 16, 1999 Aug. 17, 1999 Spot Price for Settlement Date 115.55 116.55Forward Settlement Date Aug. 15, 2000 Aug. 15, 2000 Forward Price109.217107 110.1779 Option Premium 28.333% of 29.8137% of NotionalNotional

Table 3.1.19-1 shows how the digital call option struck at 120 could, asan example, change in value with an underlying change in the yen/dollarexchange rate. The second column shows that the option is worth 28.333%or $28.333 million on a $100 million notional on Aug. 12, 1999 when theunderlying exchange rate is 115.55. The third column shows that thevalue of the option, which pays $100 million should dollar yen equal orexceed 120 at the expiration date, increases to 29.8137% or $29.8137million per $100 million when the underlying exchange rate has increasedby 1 yen to 116.55. Thus, the traditional digital call option generatesa profit of $29.81377−$28.333=$1.48077 million.

This example shows how this profit also could be realized in trading ina group of DBAR contingent claims with two successive trading periods.It is also assumed for purposes of this example that there aresufficient amounts invested, or liquidity, in both states such that theparticular trader's investment does not materially affect the returns toeach state. This is a convenient but not necessary assumption thatallows the trader to take the returns to each state “as given” withoutconcern as to how his investment will affect the closing returns for agiven trading period. Using information from Table 3.1.19-1, thefollowing closing returns for each state can be derived:

Trading Period 1:

-   -   Current trading period end date: Aug. 12, 1999    -   Underlying Event: Closing level of yen/dollar exchange rate for        Aug. 15, 2000 settlement, 4 pm EDT    -   Spot Price for Aug. 16, 1999 Settlement: 115.55

JPY/USD < 120 for JPY/USD ≧ 120 for State Aug. 15, 2000 Aug. 15, 2000Closing Returns 0.39533 2.5295

For purposes of this example, it is assumed that an illustrative traderhas $28.333 million invested in the state that the yen/dollar exchangerate equals or exceeds 120 for Aug. 15, 2000 settlement.

Trading Period 2:

-   -   Current trading period end date: Aug. 13, 1999    -   Underlying Event: Closing level of dollar/yen exchange rate for        Aug. 15, 2000 settlement, 4 pm EDT    -   Spot Price for Aug. 17, 1999 Settlement: 116.55

JPY/USD < 120 for JPY/USD ≧ 120 for State Aug. 15, 2000 Aug. 15, 2000Closing State Returns .424773 2.3542

For purposes of this example, it is also assumed that the illustrativetrader has a $70.18755 million hedging investment in the state that theyen/dollar exchange rate is less than 120 for Aug. 15, 2000 settlement.It is noted that, for the second period, the closing returns are lowerfor the state that the exchange equals or exceeds 120. This is due tothe change represented in Table 3.1.19-1 reflecting an assumed change inthe underlying market, which would make that state more likely.

The trader now has an investment in each trading period and has lockedin a profit of $1.4807 million, as shown below:

JPY/USD < 120 for JPY/USD ≧ 120 for State Aug. 15, 2000 Aug. 15, 2000Profit and Loss $70.18755 * .424773 − $−70.18755 + 28.333 * (000.000)$28.333 = $1.48077 $2.5295 = $1.48077

The illustrative trader in this example has therefore been able tolock-in or realize the profit no matter which state finally occurs. Thisprofit is identical to the profit realized in the traditional digitaloption, illustrating that systems and methods of the present inventioncan be used to provide at least daily if not more frequent realizationof profits and losses, or that risks can be hedged in virtually realtime.

In preferred embodiments, a quasi-continuous time hedge can beaccomplished, in general, by the following hedge investment, assumingthe effect of the size of the hedge trade does not materially effect thereturns:

$H = {\alpha_{t}*\frac{1 + r_{t}}{1 + r_{t + 1}^{c}}}$

where r_(t):=closing returns a state in which an investment wasoriginally made at time t

-   -   α_(t)=amount originally invested in the state at time t    -   r^(c) _(t+1):=closing returns at time t+1 to state or states        other than the state in which the original investment was made        (i.e., the so-called complement states which are all states        other than the state or states originally traded which are to be        hedged)    -   H:=the amount of the hedge investment

If H is to be invested in more than one state, then a multi-stateallocation among the constituent states can be performed using themethods and procedures described above. This expression for H allowsinvestors in DBAR contingent claims to calculate the investment amountsfor hedging transactions. In the traditional markets, such calculationsare often complex and quite difficult.

Example 3.1.20: Value Units For Investments and Payouts

As previously discussed in this specification, the units of investmentsand payouts used in embodiments of the present invention can be any unitof economic value recognized by investors, including, for example,currencies, commodities, number of shares, quantities of indices,amounts of swap transactions, or amounts of real estate. The investedamounts and payouts need not be in the same units and can comprise agroup or combination of such units, for example 25% gold, 25% barrels ofoil, and 50% Japanese Yen. The previous examples in this specificationhave generally used U.S. dollars as the value units for investments andpayouts.

This Example 3.1.20 illustrates a group of DBAR contingent claims for acommon stock in which the invested units and payouts are defined inquantities of shares.

For this example, the terms and conditions of Example 3.1.1 aregenerally used for the group of contingent claims on MSFT common stock,except for purposes of brevity, only three states are presented in thisExample 3.1.20: (0,83], (83, 88], and (88,∞]. Also in this Example3.1.20, invested amounts are in numbers of shares for each state and theexchange makes the conversion for the trader at the market priceprevailing at the time of the investment. In this example, payouts aremade according to a canonical DRF in which a trader receives a quantityof shares equal to the number of shares invested in states that did notoccur, in proportion to the ratio of number of shares the trader hasinvested in the state that did occur, divided by the total number ofshares invested in that state. An indicative distribution of traderdemand in units of number of shares is shown below, assuming that thetotal traded amount is 100,000 shares:

Return Per Share if State Occurs Amount Traded in Unit Returns in Numberof State Number of Share Shares (0, 83] 17,803 4.617 (83, 88] 72,725.37504 (88, ∞] 9,472 9.5574

If, for instance, MSFT closes at 91 at expiration, then in this examplethe third state has occurred, and a trader who had previously invested10 shares in that state would receive a payout of 10*9.5574+10=105.574shares which includes the trader's original investment. Traders who hadpreviously invested in the other two states would lose all of theirshares upon application of the canonical DRF of this example.

An important feature of investing in value units other than units ofcurrency is that the magnitude of the observed outcome may well berelevant, as well as the state that occurs based on that outcome. Forexample, if the investments in this example were made in dollars, thetrader who has a dollar invested in state (88,∞] would not care, atleast in theory, whether the final price of MSFT at the close of theobservation period were 89 or 500. However, if the value units arenumbers of shares of stock, then the magnitude of the final outcome doesmatter, since the trader receives as a payout a number of shares whichcan be converted to more dollars at a higher outcome price of $91 pershare. For instance, for a payout of 105.574 shares, these shares areworth 105.574*$91=$9,607.23 at the outcome price. Had the outcome pricebeen $125, these shares would have been worth 105.574*125=$13,196.75.

A group of DBAR contingent claims using value units of commodity havinga price can therefore possess additional features compared to groups ofDBAR contingent claims that offer fixed payouts for a state, regardlessof the magnitude of the outcome within that state. These features mayprove useful in constructing groups of DBAR contingent claims which areable to readily provide risk and return profiles similar to thoseprovided by traditional derivatives. For example, the group of DBARcontingent claims described in this example could be of great interestto traders who transact in traditional derivatives known as“asset-or-nothing digital options” and “supershares options.”

Example 3.1.21: Replication of An Arbitrary Payout Distribution

An advantage of the systems and methods of the present invention isthat, in preferred embodiments, traders can generate an arbitrarydistribution of payouts across the distribution of defined states for agroup of DBAR contingent claims. The ability to generate a customizedpayout distribution may be important to traders, since they may desireto replicate contingent claims payouts that are commonly found intraditional markets, such as those corresponding to long positions instocks, short positions in bonds, short options positions in foreignexchange, and long option straddle positions, to cite just a fewexamples. In addition, preferred embodiments of the present inventionmay enable replicated distributions of payouts which can only begenerated with difficulty and expense in traditional markets, such asthe distribution of payouts for a long position in a stock that issubject to being “stopped out” by having a market-maker sell the stockwhen it reaches a certain price below the market price. Such stop-lossorders are notoriously difficult to execute in traditional markets, andtraders are frequently not guaranteed that the execution will occurexactly at the pre-specified price.

In preferred embodiments, and as discussed above, the generation andreplication of arbitrary payout distributions across a givendistribution of states for a group of DBAR contingent claims may beachieved through the use of multi-state investments. In suchembodiments, before making an investment, traders can specify a desiredpayout for each state or some of the states in a given distribution ofstates. These payouts form a distribution of desired payouts across thedistribution of states for the group of DBAR contingent claims. Inpreferred embodiments, the distribution of desired payouts may be storedby an exchange, which may also calculate, given an existing distributionof investments across the distribution of states, (1) the total amountrequired to be invested to achieve the desired payout distribution; (2)the states into which the investment is to allocated; and (3) how muchis to be invested in each state so that the desired payout distributioncan be achieved. In preferred embodiments, this multi-state investmentis entered into a suspense account maintained by the exchange, whichreallocates the investment among the states as the amounts investedchange across the distribution of states. In preferred embodiments, asdiscussed above, a final allocation is made at the end of the tradingperiod when returns are finalized.

The discussion in this specification of multi-state investments hasincluded examples in which it has been assumed that an illustrativetrader desires a payout which is the same no matter which state occursamong the constituent states of a multi-state investment. To achievethis result, in preferred embodiments the amount invested by the traderin the multi-state investment can be allocated to the constituent statein proportion to the amounts that have otherwise been invested in therespective constituent states. In preferred embodiments, theseinvestments are reallocated using the same procedure throughout thetrading period as the relative proportion of amounts invested in theconstituent states changes.

In other preferred embodiments, a trader may make a multi-stateinvestment in which the multi-state allocation is not intended togenerate the same payout irrespective of which state among theconstituent state occurs. Rather, in such embodiments, the multi-stateinvestment may be intended to generate a payout distribution whichmatches some other desired payout distribution of the trader across thedistribution of states. Thus, the systems and methods of the presentinvention do not require amounts invested in multi-state investments tobe allocated in proportion of the amounts otherwise invested in theconstituent states of the multi-statement investment.

Notation previously developed in this specification is used to describea preferred embodiment of a method by which replication of an arbitrarydistribution of payouts can be achieved for a group of DBAR contingentclaims according to the present invention. The following additionalnotation, is also used:

A_(i,*) denotes the i-th row of the matrix A containing the investedamounts by trader i for each of the n states of the group of DBARcontingent claims In preferred embodiments, the allocation of amountsinvested in all the states which achieves the desired payouts across thedistribution of states can be calculated using, for example, thecomputer code listing in Table 1 (or functional equivalents known to oneof skill in the art), or, in the case where a trader's multi-stateinvestment is small relative to the total investments already made inthe group of DBAR contingent claims, the following approximation:A _(i,*) ^(T)=Π⁻¹ *P _(i,*) ^(T)where the −1 superscript on the matrix Π denotes a matrix inverseoperation. Thus, in these embodiments, amounts to be invested to producean arbitrary distribution payouts can approximately be found bymultiplying (a) the inverse of a diagonal matrix with the unit payoutsfor each state on the diagonal (where the unit payouts are determinedfrom the amounts invested at any given time in the trading period) and(b) a vector containing the trader's desired payouts. The equation aboveshows that the amounts to be invested in order to produce a desiredpayout distribution are a function of the desired payout distributionitself (P_(i,*)) and the amounts otherwise invested across thedistribution of states (which are used to form the matrix Π, whichcontains the payouts per unit along its diagonals and zeros along theoff-diagonals). Therefore, in preferred embodiments, the allocation ofthe amounts to be invested in each state will change if either thedesired payouts change or if the amounts otherwise invested across thedistribution change. As the amounts otherwise invested in various statescan be expected to change during the course of a trading period, inpreferred embodiments a suspense account is used to reallocate theinvested amounts, A_(i,*), in response to these changes, as describedpreviously. In preferred embodiments, at the end of the trading period afinal allocation is made using the amounts otherwise invested across thedistribution of states. The final allocation can typically be performedusing the iterative quadratic solution techniques embodied in thecomputer code listing in Table 1.

Example 3.1.21 illustrates a methodology for generating an arbitrarypayout distribution, using the event, termination criteria, the defiedstates, trading period and other relevant information, as appropriate,from Example 3.1.1, and assuming that the desired multi-state investmentis small in relation to the total amount of investments already made. InExample 3.1.1 above, illustrative investments are shown across thedistribution of states representing possible closing prices for MSFTstock on the expiration date of Aug. 19, 1999. In that example, thedistribution of investment is illustrated for Aug. 19, 1999, one dayprior to expiration, and the price of MSFT on this date is given as 85.For purposes of this Example 3.1.21, it is assumed that a trader wouldlike to invest in a group of DBAR contingent claims according to thepresent invention in a way that approximately replicates the profits andlosses that would result from owning one share of MSFT (i.e., arelatively small amount) between the prices of 80 and 90. In otherwords, it is assumed that the trader would like to replicate atraditional long position in MSFT with the restrictions that a sellorder is to be executed when MSFT reaches 80 or 90. Thus, for example,if MSFT closes at 87 on Aug. 19, 1999 the trader would expect to have $2of profit from appropriate investments in a group of DBAR contingentclaims. Using the defined states identified in Example 3.1.1, thisprofit would be approximate since the states are defined to include arange of discrete possible closing prices.

In preferred embodiments, an investment in a state receives the samereturn regardless of the actual outcome within the state. It istherefore assumed for purposes of this Example 3.1.21 that a traderwould accept an appropriate replication of the traditional profit andloss from a traditional position, subject to only “discretization”error. For purposes of this Example 3.1.21, and in preferredembodiments, it is assumed that the profit and loss corresponding to anactual outcome within a state is determined with reference to the pricewhich falls exactly in between the upper and lower bounds of the stateas measured in units of probability, i.e., the “state average.” For thisExample 3.1.21, the following desired payouts can be calculated for eachof the states the amounts to be invested in each state and the resultinginvestment amounts to achieve those payouts:

TABLE 3.1.21-1 Investment Which Generates Desired States State Average($) Desired Payout ($) Payout ($) (0,80] NA 80 0.837258 (80,80.5]80.33673 80.33673 0.699493 (80.5,81] 80.83349 80.83349 1.14091 (81,81.5]81.33029 81.33029 1.755077 (81.5,82] 81.82712 81.82712 2.549131(82,82.5] 82.32401 82.32401 3.498683 (82.5,83] 82.82094 82.820944.543112 (83,83.5] 83.31792 83.31792 5.588056 (83.5,84] 83.8149683.81496 6.512429 (84,84.5] 84.31204 84.31204 7.206157 (84.5,85]84.80918 84.80918 7.572248 (85,85.5] 85.30638 85.30638 7.555924(85.5,86] 85.80363 85.80363 7.18022 (86,86.5] 86.30094 86.30094 6.493675(86.5,87] 86.7983 86.7983 5.59628 (87,87.5] 87.29572 87.29572 4.599353(87.5,88] 87.7932 87.7932 3.611403 (88,88.5] 88.29074 88.29074 2.706645(88.5,89] 88.78834 88.78834 1.939457 (89,89.5] 89.28599 89.285991.330046 (89.5,90] 89.7837 89.7837 0.873212 (90,∞] NA 90 1.2795The far right column of Table 3.1.21-1 is the result of the matrixcomputation described above. The payouts used to construct the matrix Πfor this Example 3.1.21 are one plus the returns shown in Example 3.1.1for each state.

Pertinently the systems and methods of the present invention be used toachieve almost any arbitrary payout or return profile, e.g., a longposition, a short position, an option “straddle”, etc., whilemaintaining limited liability and the other benefits of the inventiondescribed in this specification.

As discussed above, if many traders make multi-state investments, in apreferred embodiment an iterative procedure is used to allocate all ofthe multi-state investments to their respective constituent states.Computer code, as previously described and apparent to one of skill inthe art, can be implemented to allocate each multi-state investmentamong the constituent states depending upon the distribution of amountsotherwise invested and the trader's desired payout distribution.

3.2 DBAR Portfolios

It may be desirable to combine a number of groups of DBAR contingentclaims based on different events into a single portfolio. In this way,traders can invest amounts within the distribution of defined statescorresponding to a single event as well as across the distributions ofstates corresponding to all the groups of contingent claims in theportfolio. In preferred embodiments, the payouts to the amounts investedin this fashion can therefore be a function of a relative comparison ofall the outcome states in the respective groups of DBAR contingentclaims to each other. Such a comparison may be based upon the amountinvested in each outcome state in the distribution for each group ofcontingent claims as well as other qualities, parameters orcharacteristics of the outcome state (e.g., the magnitude of change foreach security underlying the respective groups of contingent claims). Inthis way, more complex and varied payout and return profiles can beachieved using the systems and methods of the present invention. Since apreferred embodiment of a demand reallocation function (DRF) can operateon a portfolio of DBAR contingent claims, such a portfolio is referredto as a DBAR Portfolio, or DBARP. A DBARP is a preferred embodiment ofDBAR contingent claims according to the present invention based on amulti-state, multi-event DRF.

In a preferred embodiment of a DBARP involving different events relatingto different financial products, a DRF is employed in which returns foreach contingent claim in the portfolio are determined by (i) the actualmagnitude of change for each underlying financial product and (ii) howmuch has been invested in each state in the distribution. A large amountinvested in a financial product, such as a common stock, on the longside will depress the returns to defined states on the long side of acorresponding group of DBAR contingent claims. Given the inverserelationship in preferred embodiments between amounts invested in andreturns from a particular state, one advantage to a DBAR portfolio isthat it is not prone to speculative bubbles. More specifically, inpreferred embodiments a massive influx of long side trading, forexample, will increase the returns to short side states, therebyincreasing returns and attracting investment in those states.

The following notation is used to explain further preferred embodimentsof DBARP:

μ_(i) is the actual magnitude of change for financial product i W_(i) isthe amount of successful investments in financial product i L_(i) is theamount of unsuccessful investments in financial product i f is thesystem transaction fee L${{is}\mspace{14mu}{the}\mspace{14mu}{aggregate}\mspace{14mu}{losses}} = {\sum\limits_{i}\; L_{i}}$γ_(i)${{is}\mspace{14mu}{the}\mspace{14mu}{normalized}\mspace{14mu}{returns}\mspace{14mu}{for}\mspace{14mu}{successful}\mspace{14mu}{trades}} = \frac{\left| \mu_{i} \right|}{\sum\limits_{i}\left| \mu_{i} \right|}$π^(p) _(i) is the payout per value unit invested in financial product ifor a successful investment r^(p) _(i) is the return per unit investedin financial product i for a successful investment

The payout principle of a preferred embodiment of a DBARP is to returnto a successful investment a portion of aggregate losses scaled by thenormalized return for the successful investment, and to return nothingto unsuccessful investments. Thus, in a preferred embodiment a largeactual return on a relatively lightly traded financial product willbenefit from being allocated a high proportion of the unsuccessfulinvestments.

$\pi_{i}^{p} = \frac{\gamma_{i}*L}{W_{i}}$$r_{i}^{p} = {\frac{\gamma_{i}*L}{W_{i}} - 1}$

As explained below, the correlations of returns across securities isimportant in preferred embodiments to determine payouts and returns in aDBARP.

An example illustrates the operation of a DBARP according to the presentinvention. For purposes of this example, it is assumed that a portfoliocontains two stocks, IBM and MSFT (Microsoft) and that the followinginformation applies (e.g., predetermined termination criteria):

-   -   Trading start date: Sep. 1, 1999    -   Expiration date: Oct. 1, 1999    -   Current trading period start date: Sep. 1, 1999    -   Current trading period end date: Sep. 5, 1999    -   Current date: Sep. 2, 1999    -   IBM start price: 129    -   MSFT start price: 96    -   Both IBM and MSFT Ex-dividends    -   No transaction fee

In this example, states can be defined so that traders can invest forIBM and MSFT to either depreciate or appreciate over the period. It isalso assumed that the distribution of amounts invested in the variousstates is the following at the close of trading for the current tradingperiod:

Financial Product Depreciate State Appreciate State MSFT $100 million$120 million IBM  $80 million  $65 millionThe amounts invested express greater probability assessments that MSFTwill likely appreciate over the period and IBM will likely depreciate.

For purposes of this example, it is further assumed that on theexpiration date of Oct. 1, 1999, the following actual outcomes forprices are observed:

-   -   MSFT: 106 (appreciated by 10.42%)    -   IBM 127 (depreciated by 1.55%)

In this example, there is $100+$65=$165 million to distribute from theunsuccessful investments to the successful investments, and, for thesuccessful investments, the relative performance of MSFT(10/42/(10.42+1.55)=0.871) is higher than for IBM(1.55/10.42+1.55)=0.229). In a preferred embodiment, 87.1% of theavailable returns is allocated to the successful MSFT traders, with theremainder due the successful IBM traders, and with the following returnscomputed for each state:

MSFT: $120 million of successful investment produces a payout of0.871*$165 million=$143.72 million for a return to the successfultraders of

${\frac{{120\mspace{14mu} M} + {143.72\mspace{14mu} M}}{120\mspace{14mu} M} - 1} = {119.77\%}$

IBM: $80 million in successful investment produces a payoutof(1−0.871)*$165 million=$21.285 million, for a return to the successfultraders of

${\frac{{80\mspace{14mu} M} + {21.285\mspace{14mu} M}}{80\mspace{14mu} M} - 1} = {26.6\%}$The returns in this example and in preferred embodiments are a functionnot only of the amounts invested in each group of DBAR contingentclaims, but also the relative magnitude of the changes in prices for theunderlying financial products or in the values of the underlying eventsof economic performance. In this specific example, the MSFT tradersreceive higher returns since MSFT significantly outperformed IBM. Inother words, the MSFT longs were “more correct” than the IBM shorts.

The operation of a DBARP is further illustrated by assuming instead thatthe prices of both MSFT and IBM changed by the same magnitude, e.g.,MSFT went up 10%, and IBM went down 10%, but otherwise maintaining theassumptions for this example. In this scenario, $165 million of returnswould remain to distribute from the unsuccessful investments but theseare allocated equally to MSFT and IBM successful investments, or $82.5million to each. Under this scenario the returns are:

${{MSFT}:{\frac{{120\mspace{14mu} M} + {82.5\mspace{14mu} M}}{120\mspace{14mu} M} - 1}} = {68.75\%}$${{IBM}:{\frac{{80\mspace{14mu} M} + {82.5\mspace{14mu} M}}{80\mspace{14mu} M} - 1}} = {103.125\%}$The IBM returns in this scenario are 1.5 times the returns to the MSFTinvestments, since less was invested in the IBM group of DBAR contingentclaims than in the MSFT group.

This result confirms that preferred embodiments of the systems andmethods of the present invention provide incentives for traders to makelarge investments, i.e. promote liquidity, where it is needed in orderto have an aggregate amount invested sufficient to provide a fairindication of trader expectations.

The payouts in this example depend upon both the magnitude of change inthe underlying stocks as well as the correlations between such changes.A statistical estimate of these expected changes and correlations can bemade in order to compute expected returns and payouts during trading andat the close of each trading period. While making such an investment maybe somewhat more complicated that in a DBAR range derivative, asdiscussed above, it is still readily apparent to one of skill in the artfrom this specification or from practice of the invention.

The preceding example of a DBARP has been illustrated with eventscorresponding to closing prices of underlying securities. DBARPs of thepresent invention are not so limited and may be applied to any events ofeconomic significance, e.g., interest rates, economic statistics,commercial real estate rentals, etc. In addition, other types of DRFsfor use with DBARPs are apparent to one of ordinary skill in the art,based on this specification or practice of the present invention.

4 RISK CALCULATIONS

Another advantage of the groups of DBAR contingent claims according tothe present invention is the ability to provide transparent riskcalculations to traders, market risk managers, and other interestedparties. Such risks can include market risk and credit risk, which arediscussed below.

4.1 Market Risk

Market risk calculations are typically performed so that traders haveinformation regarding the probability distribution of profits and lossesapplicable to their portfolio of active trades. For all tradesassociated with a group of DBAR contingent claims, a trader might wantto know, for example, the dollar loss associated with the bottom fifthpercentile of profit and loss. The bottom fifth percentile correspondsto a loss amount which the trader knows, with a 95% statisticalconfidence, would not be exceeded. For the purposes of thisspecification, the loss amount associated with a given statisticalconfidence (e.g., 95%, 99%) for an individual investment is denoted thecapital-at-risk (“CAR”). In preferred embodiments of the presentinvention, a CAR can be computed not only for an individual investmentbut also for a plurality of investments related to for the same event orfor multiple events.

In the financial industry, there are three common methods that arecurrently employed to compute CAR: (1) Value-at-Risk (“VAR”); (2) MonteCarlo Simulation (“MCS”); and (3) Historical Simulation (“HS”).

4.1.1 Capital-At-Risk Determinations Using Value-At-Risk Techniques

VAR is a method which commonly relies upon calculations of the standarddeviations and correlations of price changes for a group of trades.These standard deviations and correlations are typically computed fromhistorical data. The standard deviation data are typically used tocompute the CAR for each trade individually.

To illustrate the use of VAR with a group of DBAR contingent claims ofthe present invention, the following assumptions are made: (i) a traderhas made a traditional purchase of a stock, say $100 of IBM; (ii) usingpreviously computed standard deviation data, it is determined that theannual standard deviation for IBM is 30%; (iii) as is commonly the case,the price changes for IBM have a normal distribution; and (iv) thepercentile of loss to be used is the bottom fifth percentile. Fromstandard normal tables, the bottom fifth percentile of loss correspondsto approximately 1.645 standard deviations, so the CAR in thisexample—that is, loss for the IBM position that would not be exceededwith 95% statistical confidence—is 30%*1.645*$100, or $49.35. A similarcalculation, using similar assumptions, has been made for a $200position in GM, and the CAR computed for GM is $65.50. If, in thisexample, the computed correlation, ξ, between the prices of IBM and GMstock is 0.5, the CAR for the portfolio containing both the IBM and GMpositions may be expressed as:

$\begin{matrix}{{CAR} = \sqrt{\begin{matrix}{\left( {1.645\alpha_{IBM}\sigma_{IBM}} \right)^{2} + \left( {1.645\alpha_{GM}\sigma_{GM}} \right)^{2} +} \\{2{\zeta 1}{.645}\alpha_{IBM}\sigma_{IBM}*1.645\alpha_{GM}\sigma_{GM}}\end{matrix}}} \\{= {\sqrt{49.35^{2} + 65.50^{2} + {2*{.5}*49.35*65.5}} = 99.79}}\end{matrix}$

where α is the investment in dollars, σis the standard deviation, and ξis the correlation.

These computations are commonly represented in matrix form as:

C is the correlation matrix of the underlying events,

w is the vector containing the CAR for each active position in theportfolio, and

w^(T) is the transpose of W.

In preferred embodiments, C is a y×y matrix, where y is the number ofactive positions in the portfolio, and where the elements of C are:

c_(ij)=1 when i=j i.e., has 1's on the diagonal, and otherwise

c_(ij)=the correlation between the ith and jth events

${CAR} = {\sqrt{w^{T}*C*w} = \sqrt{\begin{matrix}\left( 49.35 \right. & \left. 65.5 \right) & \begin{pmatrix}1 & {.5} \\{.5} & 1\end{pmatrix} & \begin{pmatrix}49.35 \\65.5\end{pmatrix}\end{matrix}}}$

In preferred embodiments, several steps implement the VAR methodologyfor a group of DBAR contingent claims of the present invention. Thesteps are first listed, and details of each step are then provided. Thesteps are as follows:

(1) beginning with a distribution of defined states for a group of DBARcontingent claims, computing the standard deviation of returns in valueunits (e.g., dollars) for each investment in a given state;

(2) performing a matrix calculation using the standard deviation ofreturns for each state and the correlation matrix of returns for thestates within the same distribution of states, to obtain the standarddeviation of returns for all investments in a group of DBAR contingentclaims;

(3) adjusting the number resulting from the computation in step (2) foreach investment so that it corresponds to the desired percentile ofloss;

(4) arranging the numbers resulting from step (3) for each distinct DBARcontingent claim in the portfolio into a vector, w, having dimensionequal to the number of distinct DBAR contingent claims;

(5) creating a correlation matrix including the correlation of each pairof the underlying events for each respective DBAR contingent claim inthe portfolio; and

(6) calculating the square root of the product of w, the correlationmatrix created in step (5), and the transpose of w.

The result is CAR using the desired percentile of loss, for all thegroups of DBAR contingent claims in the portfolio.

In preferred embodiments, the VAR methodology of steps (1)-(6) above canbe applied to an arbitrary group of DBAR contingent claims as follows.For purposes of illustrating this methodology, it is assumed that allinvestments are made in DBAR range derivatives using a canonical DRF aspreviously described. Similar analyses apply to other forms of DRFs.

In step (1), the standard deviation of returns per unit of amountinvested for each state i for each group of DBAR contingent claim iscomputed as follows:

$\sigma_{i} = {\sqrt{\frac{T}{T_{i}} - 1} = {\sqrt{\frac{\left( {1 - q_{i}} \right)}{q_{i}}} = \sqrt{r_{i}}}}$where Σ_(i) is the standard deviation of returns per unit of amountinvested in each state i, T_(i) is the total amount invested in state i;T is the sum of all amounts invested across the distribution of states;q_(i) is the implied probability of the occurrence of state i derivedfrom T and T_(i); and r_(i) is the return per unit of investment instate i. In this preferred embodiment, this standard deviation is afunction of the amount invested in each state and total amount investedacross the distribution of states, and is also equal to the square rootof the unit return for the state. If α_(i) is the amount invested instate i, α_(i)*σ_(i) is the standard deviation in units of the amountinvested (e.g., dollars) for each state i.

Step (2) computes the standard deviation for all investments in a groupof DBAR contingent claims. This step (2) begins by calculating thecorrelation between each pair of states for every possible pair withinthe same distribution of states for a group of DBAR contingent claims.For a canonical DRF, these correlations may be computed as follows:

$\rho_{i,j} = {{- \frac{\sqrt{T_{i}*T_{j}}}{\sqrt{\left( {T - T_{i}} \right)*\left( {T - T_{j}} \right)}}} = {{- \sqrt{\frac{q_{i}*q_{j}}{\left( {1 - q_{i}} \right)*\left( {1 - q_{j}} \right)}}} = {\frac{- 1}{\sqrt{r_{i}*r_{j}}} = \frac{- 1}{\sigma_{i}*\sigma_{j}}}}}$where ρ_(ij) is the correlation between state i and state j. Inpreferred embodiments, the returns to each state are negativelycorrelated since the occurrence of one state (a successful investment)precludes the occurrence of other states (unsuccessful investments). Ifthere are only two states in the distribution of states, thenT_(j)=T−T_(i) and the correlation ρ_(ij) is −1, i.e., an investment instate i is successful and in state j is not, or vice versa, if i and jare the only two states. In preferred embodiments where there are morethan two states, the correlation falls in the range between 0 and −1(the correlation is exactly 0 if and only if one of the states hasimplied probability equal to one). In step (2) of the VAR methodology,the correlation coefficients ρ_(ij) are put into a matrix C_(s) (thesubscript s indicating correlation among states for the same event)which contains a number of rows and columns equal to the number ofdefined states for the group of DBAR contingent claims. The correlationmatrix contains 1's along the diagonal, is symmetric, and the element atthe i-th row and j-th column of the matrix is equal to ρ_(ij). From step(1) above, a n×1 vector U is constructed having a dimension equal to thenumber of states n, in the group of DBAR contingent claims, with eachelement of U being equal to α_(i)*σ_(i). The standard deviation, w_(k),of returns for all investments in states within the distribution ofstates defining the kth group of DBAR contingent claims can becalculated as follows:w _(k) =√{square root over (U^(T) *C _(s) *U)}Step (3) involves adjusting the previously computed standard deviation,w_(k), for every group of DBAR contingent claims in a portfolio by anamount corresponding to a desired or acceptable percentile of loss. Forpurposes of illustration, it is assumed that investment returns have anormal distribution function; that a 95% statistical confidence forlosses is desirable; and that the standard deviations of returns foreach group of DBAR contingent claims, w_(k), can be multiplied by 1.645,i.e., the number of standard deviations in the standard normaldistribution corresponding to the bottom fifth percentile. A normaldistribution is used for illustrative purposes, and other types ofdistributions (e.g., the Student T distribution) can be used to computethe number of standard deviations corresponding to the any percentile ofinterest. As discussed above, the maximum amount that can be lost inpreferred embodiments of canonical DRF implementation of a group of DBARcontingent claims is the amount invested.

Accordingly, for this illustration the standard deviations w_(k) areadjusted to reflect the constraint that the most that can be lost is thesmaller of (a) the total amount invested and (b) the percentile loss ofinterest associated with the CAR calculation for the group of DBARcontingent claims, i.e.:

$w_{k} = {\min\left( {{1.645*w_{k}},{\sum\limits_{i = {1\ldots\; n}}\;\alpha_{i}}} \right)}$

In effect, this updates the standard deviation for each event bysubstituting for it a CAR value that reflects a multiple of the standarddeviation corresponding to an extreme loss percentile (e.g., bottomfifth) or the total invested amount, whichever is smaller.

Step (4) involves taking the adjusted w_(k), as developed in step (4)for each of m groups of DBAR contingent claims, and arranging them intoan y×1 dimensional column vector, w, each element of which containsw_(k),k=1. . . y.

Step (5) involves the development of a symmetric correlation matrix,C_(e), which has a number of rows and columns equal to the number ofgroups of DBAR contingent claims, y, in which the trader has one or moreinvestments. Correlation matrix C_(e) can be estimated from historicaldata or may be available more directly, such as the correlation matrixamong foreign exchange rates, interest rates, equity indices,commodities, and other financial products available from JP Morgan'sRiskMetrics database. Other sources of the correlation information formatrix C_(e) are known to those of skill in the art. Along the diagonalsof the correlation matrix C_(e) are 1's, and the entry at the i-th rowand j-th column of the matrix contains the correlation between the i-thand j-th events which define the i-th and j-th DBAR contingent claim forall such possible pairs among the m active groups of DBAR contingentclaims in the portfolio.

In Step (6), the CAR for the entire portfolio of m groups of DBARcontingent claims is found by performing the following matrixcomputation, using each w_(k) from step (4) arrayed into vector w andits transpose w^(T):CAR=√{square root over (w ^(T) *C _(e) *w)}This CAR value for the portfolio of groups of DBAR contingent claims isan amount of loss which will not be exceeded with the associatedstatistical confidence used in Steps (1)-(6) above (e.g., in thisillustration, 95%).

Example 4.1.1-1: VAR-based CAR Calculation

An example further illustrates the calculation of a VAR-based CAR for aportfolio containing two groups of DBAR range derivative contingentclaims (i.e., y=2) with a canonical DRF on two common stocks, IBM andGM. For this example, the following assumptions are made: (i) for eachof the two groups of DBAR contingent claims, the relevant underlyingevent upon which the states are defined is the respective closing priceof each stock one month forward; (ii) there are only three statesdefined for each event: “low”, “medium”, and “high,” corresponding toranges of possible closing prices on that date; (iii) the posted returnsfor IBM and GM respectively for the three respective states are, in U.S.dollars, (4, 0.6667, 4) and (2.333, 1.5, 2.333); (iv) the exchange feeis zero; (v) for the IBM group of contingent claims, the trader has onedollar invested in the state “low”, three dollars invested in the state“medium,” and two dollars invested in the state “high”; (vi) for the GMgroup of contingent claims, the trader has a single investment in theamount of one dollar in the state “medium”; (vii) the desired oracceptable percentile of loss in the fifth percentile, assuming a normaldistribution; and (viii) the estimated correlation of the price changesof IBM and GM is 0.5 across the distribution of states for each stock.

Steps (1)-(6), described above, are used to implement VAR in order tocompute CAR for this example. From Step (1), the standard deviations ofstate returns per unit of amount invested in each state for the IBM andGM groups of contingent claims are, respectively, (2, 0.8165, 2) and(1.5274, 1.225, 1.5274). In further accordance with Step (1) above, theamount invested in each state in the respective group of contingentclaims, α_(i); is multiplied by the previously calculated standarddeviation of state returns per investment, σ_(i), so that the standarddeviation of returns per state in dollars for each claim equals, for theIBM group: (2, 2.4495, 4) and, for the GM group, (0,1.225, 0).

In accordance with Step (2) above, for each of the two groups of DBARcontingent claims in this example, a correlation matrix between any pairof states, C_(s), is constructed, as follows:

${C_{s}^{IBM} = \begin{matrix}1 & {- {.6124}} & {- {.25}} \\{- {.6124}} & 1 & {- {.6124}} \\{- {.25}} & {- {.6124}} & 1\end{matrix}}$ $C_{s}^{GM} = \begin{matrix}1 & {- {.5345}} & {- {.4286}} \\{- {.5345}} & 1 & {- {.5345}} \\{- {.4286}} & {- {.5345}} & 1\end{matrix}$where the left matrix is the correlation between each pair of statereturns for the IBM group of contingent claims and the right matrix isthe corresponding matrix for the GM group of contingent claims.

Also according to step (2) above, for each of the two groups ofcontingent claims, the standard deviation of returns per state indollars, α_(iσ) _(i), for each investment in this example can bearranged in a vector with dimension equal to three (i.e., the number ofstates):

${U_{IBM} = \begin{matrix}2 \\2.4495 \\4\end{matrix}}$ ${U_{GM} = \begin{matrix}0 \\1.225 \\0\end{matrix}}$where the vector on the left contains the standard deviation in dollarsof returns per state for the IBM group of contingent claims, and thevector on the right contains the corresponding information for the GMgroup of contingent claims. Further in accordance with Step (2) above, amatrix calculation can be performed to compute the total standarddeviation for all investments in each of the two groups of contingentclaims, respectively:w ₁ =√{square root over (U_(IBM) ^(T) *C _(s) ^(IBM) *U _(IBM))}=2 w ₂=√{square root over (U_(GM) ^(T) *C _(s) ^(GM) *U _(GM))}=1.225where the quantity on the left is the standard deviation for allinvestments in the distribution of the IBM group of contingent claims,and the quantity on the right is the corresponding standard deviationfor the GM group of contingent claims.

In accordance with step (3) above, w₁, and w2 are adjusted bymultiplying each by 1.645 (corresponding to a CAR loss percentile of thebottom fifth percentile assuming a normal distribution) and then takingthe lower of (a) that resulting value and (b) the maximum amount thatcan be lost, i.e., the amount invested in all states for each group ofcontingent claims:w ₁=min(2*1.645,6)=3.29 w ₂=min(2*1.225,1)=1where the left quantity is the adjusted standard deviation of returnsfor all investments across the distribution of the IBM group ofcontingent claims, and the right quantity is the corresponding amountinvested in the GM group of contingent claims. These two quantities, w₁and w₂ are the CAR values for the individual groups of DBAR contingentclaims respectively, corresponding to a statistical confidence of 95%.In other words, if the normal distribution assumptions that have beenmade with respect to the state returns are valid, then a trader, forexample, could be 95% confident that losses on the IBM groups ofcontingent claims would not exceed $3.29.

Proceeding now with Step (4) in the VAR process described above, thequantities w₁ and w₂ are placed into a vector which has a dimension oftwo, equal to the number of groups of DBAR contingent claims in theillustrative trader's portfolio:

$w = \begin{matrix}3.29 \\1\end{matrix}$

According to Step (5), a correlation matrix C_(e) with two rows and twocolumns, is either estimated from historical data or obtained from someother source (e.g., RiskMetrics), as known to one of skill in the art.Consistent with the assumption for this illustration that the estimatedcorrelation between the price changes of IBM and GM is 0.5, thecorrelation matrix for the underlying events is as follows:

$C_{e} = {\begin{matrix}1 \\{.5}\end{matrix}\begin{matrix}{.5} \\1\end{matrix}}$

Proceeding with Step (6), a matrix multiplication is performed by pre-and post-multiplying C_(e) by the transpose of w and by w, and takingthe square root of the resulting product:CAR=√{square root over (w ^(T) *C _(e) *w)}=3.8877This means that for the portfolio in this example, comprising the threeinvestments in the IBM group of contingent claims and the singleinvestment in the GM group of contingent claims, the trader can have a95% statistical confidence he will not have losses in excess of $3.89.

4.1.2 Capital-At-Risk Determinations Using Monte Carlo SimulationTechniques

Monte Carlo Simulation (MCS) is another methodology that is frequentlyused in the financial industry to compute CAR. MCS is frequently used tosimulate many representative scenarios for a given group of financialproducts, compute profits and losses for each representative scenario,and then analyze the resulting distribution of scenario profits andlosses. For example, the bottom fifth percentile of the distribution ofthe scenario profits and losses would correspond to a loss for which atrader could have a 95% confidence that it would not be exceeded. In apreferred embodiment, the MCS methodology can be adapted for thecomputation of CAR for a portfolio of DBAR contingent claims as follows.

Step (1) of the MCS methodology involves estimating the statisticaldistribution for the events underlying the DBAR contingent claims usingconventional econometric techniques, such as GARCH. If the portfoliobeing analyzed has more than one group of DBAR contingent claim, thenthe distribution estimated will be what is commonly known as amultivariate statistical distribution which describes the statisticalrelationship between and among the events in the portfolio. For example,if the events are underlying closing prices for stocks and stock pricechanges have a normal distribution, then the estimated statisticaldistribution would be a multivariate normal distribution containingparameters relevant for the expected price change for each stock, itsstandard deviation, and correlations between every pair of stocks in theportfolio. Multivariate statistical distribution is typically estimatedfrom historical time series data on the underlying events (e.g., historyof prices for stocks) using conventional econometric techniques.

Step (2) of the MCS methodology involves using the estimated statisticaldistribution of Step (1) in order to simulate the representativescenarios. Such simulations can be performed using simulation methodscontained in such reference works as Numerical Recipes in C or by usingsimulation software such as @Risk package available from Palisade, orusing other methods known to one of skill in the art. For each simulatedscenario, the DRF of each group of DBAR contingent claims in theportfolio determines the payouts and profits and losses on the portfoliocomputed.

Using the above two stock example involving GM and IBM used above todemonstrate VAR techniques for calculating CAR, a scenario simulated byMCS techniques might be “High” for IBM and “Low” for GM, in which casethe trader with the above positions would have a four dollar profit forthe IBM contingent claim and a one dollar loss for the GM contingentclaim, and a total profit of three dollars. In step (2), many suchscenarios are generated so that a resulting distribution of profit andloss is obtained. The resulting profits and losses can be arranged intoascending order so that, for example, percentiles corresponding to anygiven profit and loss number can be computed. A bottom fifth percentile,for example, would correspond to a loss for which the trader could be95% confident would not be exceeded, provided that enough scenarios havebeen generated to provide an adequate representative sample. This numbercould be used as the CAR value computed using MCS for a group of DBARcontingent claims. Additionally, statistics such as average profit orloss, standard deviation, skewness, kurtosis and other similarquantities can be computed from the generated profit and lossdistribution, as known by one of skill in the art.

4.1.3 Capital-At-Risk Determination Using Historical SimulationTechniques

Historical Simulation (HS) is another method used to compute CAR values.HS is comparable to that of MCS in that it relies upon the use ofrepresentative scenarios in order to compute a distribution of profitand loss for a portfolio. Rather than rely upon simulated scenarios froman estimated probability distribution, however, HS uses historical datafor the scenarios. In a preferred embodiment, HS can be adapted to applyto a portfolio of DBAR contingent claims as follows.

Step (1) involves obtaining, for each of the underlying eventscorresponding to each group of DBAR contingent claims, a historical timeseries of outcomes for the events. For example, if the events are stockclosing prices, time series of closing prices for each stock can beobtained from a historical database such as those available fromBloomberg, Reuters, or Datastream or other data sources known to someoneof skill in the art.

Step (2) involves using each observation in the historical data fromStep (1) to compute payouts using the DRF for each group of DBARcontingent claims in the portfolio. From the payouts for each group foreach historical observation, a portfolio profit and loss can becomputed. This results in a distribution of profits and lossescorresponding to the historical scenarios, i.e., the profit and lossthat would have been obtained had the trader held the portfoliothroughout the period covered by the historical data sample.

Step (3) involves arranging the values for profit and loss from thedistribution of profit and loss computed in Step (2) in ascending order.A profit and loss can therefore be computed corresponding to anypercentile in the distribution so arranged, so that, for example, a CARvalue corresponding to a statistical confidence of 95% can be computedby reference to the bottom fifth percentile.

4.2 Credit Risk

In preferred embodiments of the present invention, a trader may makeinvestments in a group of DBAR contingent claims using a margin loan. Inpreferred embodiments, credit risk may be measured by estimating theamount of possible loss that other traders in the group of contingentclaims could suffer owing to the inability of a given trader to repay amargin loan. For example, a trader may have invested $1 in a given statefor a group of DBAR contingent claims with $0.50 of margin. Assuming acanonical DRF for this example, if the state later fails to occur, theDRF collects $1 from the trader (ignoring interest) which would requirerepayment of the margin loan. As the trader may be unable to repay themargin loan at the required time, the traders with successful trades maypotentially not be able to receive the full amounts owing them under theDRF, and may therefore receive payouts lower than those indicated by thefinalized returns for a given trading period for the group of contingentclaims. Alternatively, the risk of such possible losses due to creditrisk may be insured, with the cost of such insurance either borne by theexchange or passed on to the traders. One advantage of the system andmethod of the present invention is that, in preferred embodiments, theamount of credit risk associated with a group of contingent claims canreadily be calculated.

In preferred embodiments, the calculation of credit risk for a portfolioof groups of DBAR contingent claims involves computing acredit-capital-at-risk (CCAR) figure in a manner analogous to thecomputation of CAR for market risk, as described above.

The computation of CCAR involves the use of data related to the amountof margin used by each trader for each investment in each state for eachgroup of contingent claims in the portfolio, data related to theprobability of each trader defaulting on the margin loan (which cantypically be obtained from data made available by credit ratingagencies, such as Standard and Poors, and data related to thecorrelation of changes in credit ratings or default probabilities forevery pair of traders (which can be obtained, for example, from J PMorgan's CreditMetrics database).

In preferred embodiments, CCAR computations can be made with varyinglevels of accuracy and reliability. For example, a calculation of CCARwhich is substantially accurate but could be improved with more data andcomputational effort may nevertheless be adequate, depending upon thegroup of contingent claims and the desires of traders for credit riskrelated information. The VAR methodology, for example, can be adapted tothe computation of CCAR for a group of DBAR contingent claims, althoughit is also possible to use MCS and HS related techniques for suchcomputations. The steps that can be used in a preferred embodiment tocompute CCAR using VAR-based, MCS-based, and HS-based methods aredescribed below.

4.2.1 CCAR Method for DBAR Contingent Claims Using the VAR-basedMethodology

Step (i) of the VAR-based CCAR methodology involves obtaining, for eachtrader in a group of DBAR contingent claims, the amount of margin usedto make each trade.

Step (ii) involves obtaining data related to the probability of defaultfor each trader who has invested in the groups of DBAR contingentclaims. Default probabilities can be obtained from credit ratingagencies, from the J P Morgan CreditMetrics database, or from othersources as known to one of skill in the art. In addition to defaultprobabilities, data related to the amount recoverable upon default canbe obtained. For example, an AA-rated trader with $1 in margin loans maybe able to repay $0.80 dollars in the event of default.

Step (iii) involves scaling the standard deviation of returns in unitsof the invested amounts. This scaling step is described in step (1) ofthe VAR methodology described above for estimating market risk. Thestandard deviation of each return, determined according to Step (1) ofthe VAR methodology previously described, is scaled by (a) thepercentage of margin for each investment; (b) the probability of defaultfor the trader; and (c) the percentage not recoverable in the event ofdefault.

Step (iv) of this VAR-based CCAR methodology involves taking from step(iii) the scaled values for each state for each investment andperforming the matrix calculation described in Step (2) above for theVAR methodology for estimating market risk, as described above. In otherwords, the standard deviations of returns in units of invested amountswhich have been scaled as described in Step (iii) of this CCARmethodology are weighted according to the correlation between eachpossible pair of states (matrix C_(s), as described above). Theresulting number is a credit-adjusted standard deviation of returns inunits of the invested amounts for each trader for each investment on theportfolio of groups of DBAR contingent claims. For a group of DBARcontingent claims, the standard deviations of returns that have beenscaled in this fashion are arranged into a vector whose dimension equalsthe number of traders.

Step (v) of this VAR-based CCAR methodology involves performing a matrixcomputation, similar to that performed in Step (5) of the VARmethodology for CAR described above. In this computation, the vector ofcredit-scaled standard deviations of returns from step (iv) are used topre- and post-multiply a correlation matrix with rows and columns equalto the number of traders, with 1's along the diagonal, and with theentry at row i and column j containing the statistical correlation ofchanges in credit ratings described above. The square root of theresulting matrix multiplication is an approximation of the standarddeviation of losses, due to default, for all the traders in a group ofDBAR contingent claims. This value can be scaled by a number of standarddeviations corresponding to a statistical confidence of thecredit-related loss not to be exceeded, as discussed above.

In a preferred embodiment, any given trader may be omitted from a CCARcalculation. The result is the CCAR facing the given trader due to thecredit risk posed by other traders who have invested in a group of DBARcontingent claims. This computation can be made for all groups of DBARcontingent claims in which a trader has a position, and the resultingnumber can be weighted by the correlation matrix for the underlyingevents, C_(e), as described in Step (5) for the VAR-based CARcalculation. The result corresponds to the risk of loss posed by thepossible defaults of other traders across all the states of all thegroups of DBAR contingent claims in a trader's portfolio.

4.2.2 CCAR Method for DBAR Contingent Claims Using the Monte CarloSimulation (MCS) Methodology

As described above, MCS methods are typically used to simulaterepresentative scenarios for a given group of financial products,compute profits and losses for each representative scenario, thenanalyze the resulting distribution of scenario profits and losses. Thescenarios are designed to be representative in that they are supposed tobe based, for instance, on statistical distributions which have beenestimated, typically using econometric time series techniques, to have agreat degree of relevance for the future behavior of the financialproducts. A preferred embodiment of MCS methods to estimate CCAR for aportfolio of DBAR contingent claims of the present invention, involvestwo steps, as described below.

Step (i) of the MCS methodology is to estimate a statisticaldistribution of the events of interest. In computing CCAR for a group ofDBAR contingent claims, the events of interest may be both the primaryevents underlying the groups of DBAR contingent claims, including eventsthat may be fitted to multivariate statistical distributions to computeCAR as described above, as well as the events related to the default ofthe other investors in the groups of DBAR contingent claims. Thus, in apreferred embodiment, the multivariate statistical distribution to beestimated relates to the market events (e.g., stock price changes,changes in interest rates) underlying the groups of DBAR contingentclaims being analyzed as well as the event that the investors in thosegroups of DBAR contingent claims, grouped by credit rating orclassification will be unable to repay margin loans for losinginvestments.

For example, a multivariate statistical distribution to be estimatedmight assume that changes in the market events and credit ratings orclassifications are jointly normally distributed. Estimating such adistribution would thus entail estimating, for example, the mean changesin the underlying market events (e.g., expected changes in interestrates until the expiration date), the mean changes in credit ratingsexpected until expiration, the standard deviation for each market eventand credit rating change, and a correlation matrix containing all of thepairwise correlations between every pair of events, including market andcredit event pairs. Thus, a preferred embodiment of MCS methodology atit applies to CCAR estimation for groups of DBAR contingent claims ofthe present invention typically requires some estimation as to thestatistical correlation between market events (e.g., the change in theprice of a stock issue) and credit events (e.g., whether an investorrated A− by Standard and Poors is more likely to default or bedowngraded if the price of a stock issue goes down rather than up).

It is sometimes difficult to estimate the statistical correlationsbetween market-related events such as changes in stock prices andinterest rates, on the one hand, and credit-related events such ascounterparty downgrades and defaults, on the other hand. Thesedifficulties can arise due to the relative infrequency of creditdowngrades and defaults. The infrequency of such credit-related eventsmay mean that statistical estimates used for MCS simulation can only besupported with low statistical confidence. In such cases, assumptionscan be employed regarding the statistical correlations between themarket and credit-related events. For example, it is not uncommon toemploy sensitivity analysis with regard to such correlations, i.e., toassume a given correlation between market and credit-related events andthen vary the assumption over the entire range of correlations from −1to 1 to determine the effect on the overall CCAR.

A preferred approach to estimating correlation between events is to usea source of data with regard to credit-related events which does nottypically suffer from a lack of statistical frequency. Two methods canbe used in this preferred approach. First, data can be obtained whichprovide greater statistical confidence with regard to credit-relatedevents. For example, expected default frequency data can be purchasedfrom such companies as KMV Corporation. These data supply probabilitiesof default for various parties which can be updated as frequently asdaily. Second, more frequently observed default probabilities can beestimated from market interest rates. For example, data providers suchas Bloomberg and Reuters typically provide information on the additionalyield investors require for investments in bonds of varying creditratings, e.g., AAA, AA, A, A−. Other methods are readily available toone skilled in the art to provide estimates regarding defaultprobabilities for various entities. Such estimates can be made asfrequently as daily so that it is possible to have greater statisticalconfidence in the parameters typically needed for MCS, such as thecorrelation between changes in default probabilities and changes instock prices, interest rates, and exchange rates.

The estimation of such correlations is illustrated assuming two groupsof DBAR contingent claims of interest, where one group is based upon theclosing price of IBM stock in three months, and the other group is basedupon the closing yield of the 30-year U.S. Treasury bond in threemonths. In this illustration, it is also assumed that the counterpartieswho have made investments on margin in each of the groups can be dividedinto five distinct credit rating classes. Data on the daily changes inthe price of IBM and the bond yield may be readily obtained from suchsources as Reuters or Bloomberg. Frequently changing data on theexpected default probability of investors can be obtained, for example,from KMV Corporation, or estimated from interest rate data as describedabove. As the default probability ranges between 0 and 1, a statisticaldistribution confined to this interval is chosen for purposes of thisillustration. For example, for purposes of this illustration, it can beassumed that the expected default probability of the investors follows alogistic distribution and that the joint distribution of changes in IBMstock and the 30-year bond yield follows a bivariate normaldistribution. The parameters for the logistic distribution and thebivariate normal distribution can be estimated using econometrictechniques known to one skilled in the art.

Step (ii) of a MCS technique, as it may be applied to estimating CCARfor groups of DBAR contingent claims, involves the use of themultivariate statistical distributions estimated in Step (i) above inorder to simulate the representative scenarios. As described above, suchsimulations can be performed using methods and software readilyavailable and known to those of skill in the art. For each simulatedscenario, the simulated default rate can be multiplied by the amount oflosses an investor faces based upon the simulated market changes and themargin, if any, the investor has used to make losing investments. Theproduct represents an estimated loss rate due to investor defaults. Manysuch scenarios can be generated so that a resulting distribution ofcredit-related expected losses can be obtained. The average value of thedistribution is the mean loss. The lowest value of the top fifthpercentile of the distribution, for example, would correspond to a lossfor which a given trader could be 95% confident would not be exceeded,provided that enough scenarios have been generated to provide astatistically meaningful sample. In preferred embodiments, the selectedvalue in the distribution, corresponding to a desired or adequateconfidence level, is used as the CCAR for the groups of DBAR contingentclaims being analyzed.

4.2.3 CCAR Method for DBAR Contingent Claims Using the HistoricalSimulation (HS) Methodology

As described above, Historical Simulation (HS) is comparable to MCS forestimating CCAR in that HS relies on representative scenarios in orderto compute a distribution of profit and loss for a portfolio of groupsof DBAR contingent claim investments. Rather than relying on simulatedscenarios from an estimated multivariate statistical distribution,however, HS uses historical data for the scenarios. In a preferredembodiment, HS methodology for calculating CCAR for groups of DBARcontingent claims uses three steps, described below.

Step (i) involves obtaining the same data for the market-related eventsas described above in the context of CAR. In addition, to use HS toestimate CCAR, historical time series data are also used forcredit-related events such as downgrades and defaults. As such data aretypically rare, methods described above can be used to obtain morefrequently observed data related to credit events. For example, in apreferred embodiment, frequently-observed data on expected defaultprobabilities can be obtained from KMV Corporation. Other means forobtaining such data are known to those of skill in the art.

Step (ii) involves using each observation in the historical data fromthe previous step (i) to compute payouts using the DRF for each group ofDBAR contingent claims being analyzed. The amount of margin to be repaidfor the losing trades can then be multiplied by the expected defaultprobability to use HS to estimate CCAR, so that an expected loss numbercan be obtained for each investor for each group of contingent claims.These losses can be summed across the investment by each trader so that,for each historical observation data point, an expected loss amount dueto default can be attributed to each trader. The loss amounts can alsobe summed across all the investors so that a total expected loss amountcan be obtained for all of the investors for each historical data point.

Step (iii) involves arranging, in ascending order, the values of lossamounts summed across the investors for each data point from theprevious step (iii). An expected loss amount due to credit-relatedevents can therefore be computed corresponding to any percentile in thedistribution so arranged. For example, a CCAR value corresponding to a95% statistical confidence level can be computed by reference to 95^(th)percentile of the loss distribution.

5 LIQUIDITY AND PRICE/QUANTITY RELATIONSHIPS

In the trading of contingent claims, whether in traditional markets orusing groups of DBAR contingent claims of the present invention, it isfrequently useful to distinguish between the fundamental value of theclaim, on the one hand, as determined by market expectations,information, risk aversion and financial holdings of traders, and thedeviations from such value due to liquidity variations, on the otherhand. For example, the fair fundamental value in the traditional swapmarket for a five-year UK swap (i.e., swapping fixed interest forfloating rate payments based on UK LIBOR rates) might be 6.79% with a 2basis point bid/offer (i.e., 6.77% receive, 6.81% pay). A large traderwho takes the market's fundamental mid-market valuation of 6.79% ascorrect or fair might want to trade a swap for a large amount, such as750 million pounds. In light of likely liquidity available according tocurrent standards of the traditional market, the large amount of thetransaction could reduce the likely offered rate to 6.70%, which is afull 7 basis points lower than the average offer (which is probablyapplicable to offers of no more than 100 million pounds) and 9 basispoints away from the fair mid-market value.

The difference in value between a trader's position at the fair ormid-market value and the value at which the trade can actually becompleted, i.e,. either the bid or offer, is usually called theliquidity charge. For the illustrative five-year UK swap, a 1 basispoint liquidity charge is approximately equal to 0.04% of the amounttraded, so that a liquidity charge of 9 basis points equalsapproximately 2.7 million pounds. If no new information or otherfundamental shocks intrude into or “hit” the market, this liquiditycharge to the trader is almost always a permanent transaction charge forliquidity—one that also must be borne when the trader decides toliquidate the large position. Additionally, there is no currentlyreliable way to predict, in the traditional markets, how therelationship between price and quantity may deviate from the posted bidand offers, which are usually applicable only to limited orrepresentative amounts. Price and quantity relationships can be highlyvariable, therefore, due to liquidity variations. Those relationshipscan also be non-linear. For instance, it may cost more than twice asmuch, in terms of a bid/offer spread, to trade a second position that isonly twice as large as a first position.

From the point of view of liquidity and transactions costs, groups ofDBAR contingent claims of the present invention offer advantagescompared to traditional markets. In preferred embodiments, therelationship between price (or returns) and quantity invested (i.e.,demanded) is determined mathematically by a DRF. In a preferredembodiment using a canonical DRF, the implied probability q_(i) for eachstate i increases, at a decreasing rate, with the amount invested inthat state:

$q_{1} = \frac{T_{i}}{T}$$\frac{\partial q_{i}}{\partial T_{i}} = \frac{T - T_{i}}{T^{2}}$$\frac{\partial^{2}q_{i}}{\partial T_{i}^{2}} = {{- 2}*\frac{T - T_{i}}{T^{3}}}$$\frac{\partial q_{i}}{\partial T_{j,{j \neq i}}} = {{- \frac{T_{i}}{T^{2}}} = {- \frac{q_{i}}{T}}}$where T is the total amount invested across all the states of the groupof DBAR contingent claims and T_(i) is the amount invested in the statei. As a given the amount gets very large, the implied probability ofthat state asymptotically approaches one. The last expressionimmediately above shows that there is a transparent relationship,available to all traders, between implied probabilities and the amountinvested in states other than a given state i. The expression shows thatthis relationship is negative, i.e., as amounts invested in other statesincrease, the implied probability for the given state i decreases. Sinceadding states other than the given state is equivalent to selling thegiven state in the market, the expression for

$\frac{\partial q_{i}}{\partial T_{j,{j \neq i}}}$above shows how, in a preferred embodiment, the implied probability forthe given state changes as a quantity for that state is up for sale,i.e.,what the market's “bid” is for the quantity up for sale. Theexpression for

$\frac{\partial q_{i}}{\partial T_{i}}$above shows, in a preferred embodiment, how the probability for thegiven state changes when a given quantity is demanded or desired to bepurchased, i.e.,what the market's “offer” price is to purchasers of thedesired quantity.

In a preferred embodiment, for each set of quantities invested in thedefined states of a group of DBAR contingent claims, a set of bid andoffer curves is available as a function of the amount invested.

In the groups of DBAR contingent claims of the present invention, thereare no bids or offers per se. The mathematical relationships above areprovided to illustrate how the systems and methods of the presentinvention can, in the absence of actual bid/offer relationships, providegroups of DBAR contingent claims with some of the functionality ofbid/offer relationships.

Economists usually prefer to deal with demand and cross-demandelasticities, which are the percentage changes in prices due topercentage changes in quantity demanded for a given good (demandelasticity) or its substitute (cross-demand elasticity). In preferredembodiments of the systems and methods of the present invention, andusing the notation developed above,

${\frac{\Delta\; q_{i}}{q_{i}}/\frac{\Delta\; T_{i}}{T_{i}}} = {1 - q_{i}}$${\frac{\Delta\; q_{i}}{q_{i}}/\frac{\Delta\; T_{j}}{T_{j}}} = {- q_{j}}$

The first of the expressions immediately above shows that smallpercentage changes in the amount invested in state i have a decreasingpercentage effect on the implied probability for state i, as state ibecomes more likely (i.e., as q_(i) increases to 1). The secondexpression immediately above shows that a percentage change in theamount invested in a state j other than state i will decrease theimplied probability for state i in proportion to the implied probabilityfor the other state j.

In preferred embodiments, in order to effectively “sell” a state,traders need to invest or “buy” complement states, i.e., states otherthan the one they wish to “sell.” Thus, in a preferred embodimentinvolving a group of DBAR claims with two states, a “seller”of state 1will “buy” state 2, and vice versa In order to “sell” state 1, state 2needs to be “bought” in proportion to the ratio of the amount investedin state 2 to the amount invested in state 1. In a state distributionwhich has more than two states, the “complement” for a given state to be“sold” are all of the other states for the group of DBAR contingentclaims. Thus, “selling” one state involves “buying” a multi-stateinvestment, as described above, for the complement states.

Viewed from this perspective, an implied offer is the resulting effecton implied probabilities from making a small investment in a particularstate. Also from this perspective, an implied bid is the effect onimplied probabilities from making a small multi-state investment incomplement states. For a given state in a preferred embodiment of agroup of DBAR contingent claims, the effect of an invested amount onimplied probabilities can be stated as follows:

${{Implied}\mspace{14mu}{``{Bid}"}} = {q_{i} - {\frac{\left( {1 - q_{i}} \right)}{T}*\Delta\; T_{i}}}$${{{Implied}\mspace{14mu}{``{Offer}"}} = {q_{i} + {q_{i}*\left( {\frac{1}{T_{i}} - \frac{1}{T}} \right)*\Delta\; T_{i}}}}\mspace{14mu}$where ΔT_(i) (considered here to be small enough for a first-orderapproximation) is the amount invested for the “bid” or “offer.” Theseexpressions for implied “bid” and implied “offer” can be used forapproximate computations. The expressions indicate how possibleliquidity effects within a group of DBAR contingent claims can be castin terms familiar in traditional markets. In the traditional markets,however, there is no ready way to compute such quantities for any givenmarket.

The full liquidity effect—or liquidity response function—between twostates in a group of DBAR contingent claims can be expressed asfunctions of the amounts invested in a given state, T_(i), and amountsinvested in the complement states, denoted T^(c) _(i), as follows:

Implied “Bid” Demand Response${q_{i}^{B}\left( {\Delta\; T_{i}} \right)} = \frac{T_{i}}{T_{i} + T_{i}^{c} + {\Delta\; T_{i}*\left( \frac{T_{i}^{c}}{T_{i} - {\Delta\; T_{i}}} \right)}}$Implied “Offer” Demand Response${q_{i}^{O}\left( {\Delta\; T_{i}} \right)} = \frac{T_{i} + {\Delta\; T_{i}}}{T_{i} + T_{i}^{c} + {\Delta\; T_{i}}}$The implied “bid” demand response function shows the effect on theimplied state probability of an investment made to hedge an investmentof size ΔT_(i). The size of the hedge investment in the complementstates is proportional to the ratio of investments in the complementstates to the amount of investments in the state or states to be hedged,excluding the investment to be hedged (i.e., the third term in thedenominator). The implied “offer” demand response function above showsthe effect on the implied state probability from an incrementalinvestment of size ΔT_(i) in a particular defined state.

In preferred embodiments of systems and methods of the presentinvention, only the finalized returns for a given trading period areapplicable for computing payouts for a group of DBAR contingent claims.Thus, in preferred embodiments, unless the effect of a trade amount onreturns is permanent, i.e., persists through the end of a tradingperiod, a group of DBAR contingent claims imposes no permanent liquiditycharge, as the traditional markets typically do. Accordingly, inpreferred embodiments, traders can readily calculate the effect onreturns from investments in the DBAR contingent claims, and unless thesecalculated effects are permanent, they will not affect closing returnsand can, therefore, be ignored in appropriate circumstances. In otherwords, investing in a preferred embodiment of a group of DBAR contingentclaims does not impose a permanent liquidity charge on traders forexiting and entering the market, as the traditional markets typicallydo.

The effect of a large investment may, of course, move intra-tradingperiod returns in a group of DBAR contingent claims as indicated by theprevious calculations. In preferred embodiments, these effects couldwell be counteracted by subsequent investments that move the market backto fair value (in the absence of any change in the fundamental or fairvalue). In traditional markets, by contrast, there is usually a “tollbooth” effect in the sense that a toll or change is usually exactedevery time a trader enters and exits the market. This toll is largerwhen there is less “traffic” or liquidity and represents a permanentloss to the trader. By contrast, other than an exchange fee, inpreferred embodiments of groups of DBAR contingent claims, there is nosuch permanent liquidity tax or toll for market entry or exit.

Liquidity effects may be permanent from investments in a group of DBARcontingent claims if a trader is attempting to make a relatively verylarge investment near the end of a trading period, such that the marketmay not have sufficient time to adjust back to fair value. Thus, inpreferred embodiments, there should be an inherent incentive not to holdback large investments until the end of the trading period, therebyproviding incentives to make large investments earlier, which isbeneficial overall to liquidity and adjustment of returns. Nonetheless,a trader can readily calculate the effects on returns to a investmentwhich the trader thinks might be permanent (e.g., at the end of thetrading period), due to the effect on the market from a large investmentamount.

For example, in the two period hedging example (Example 3.1.19) above,it was assumed that the illustrated trader's investments had no materialeffect on the posted returns, in other words, that this trader was a“price taker.” The formula for the hedge trade H in the second period ofthat example above reflects this assumption. The following equivalentexpression for H takes account of the possibly permanent effect that alarge trade investment might have on the closing returns (because, forexample, the investment is made very close to the end of the tradingperiod):

$H = \frac{P_{t} - T_{t + 1} + \sqrt{T_{t + 1}^{2} - {2*T_{t + 1}*P_{t}} + P_{t}^{2} + {4*P_{t}*T_{t + 1}^{c}}}}{2}$whereP _(t)=α_(t)*(1+r _(t))in the notation used in Example 3.1.19, above, and T_(t+1) is the totalamount invested in period t+1 and T^(c) _(t+1) is the amount invested inthe complement state in period t+1. The expression for H is thequadratic solution which generates a desired payout, as described abovebut using the present notation. For example, if $1 billion is the totalamount, T, invested in trading period 2, then, according to the aboveexpressions, the hedge trade investment assuming a permanent effect onreturns is $70.435 million compared to $70.18755 million in Example3.1.19. The amount of profit and loss locked-in due to the new hedge is$1.232 million, compared to $1.48077 in Example 3.1.19. The differencerepresents the liquidity effect, which even in the example where theinvested notional is 10% of the total amount invested, is quitereasonable in a market for groups of DBAR contingent claims. There is noready way to estimate or calculate such liquidity effects in traditionalmarkets.

6 DETAILED DESCRIPTION OF THE DRAWINGS

Referring now to the drawings, similar components appearing in differentdrawings are identified by the same reference numbers.

FIGS. 1 and 2 show schematically a preferred embodiment of a networkarchitecture for a DBAR contingent claims exchange. As depicted in FIG.1 and FIG. 2, the architecture conforms to a distributed Internet-basedarchitecture using object oriented principles useful in carrying out themethods of the present invention.

In FIG. 1, a central controller 100 has a plurality software andhardware components and is embodied as a mainframe computer or aplurality of workstations. The central controller 100 is preferablylocated in a facility that has back-up power, disaster-recoverycapabilities, and other similar infrastructure, and is connected viatelecommunications links 110 with computers and devices 160, 170, 180,190, and 200 of traders and investors in groups of DBAR contingentclaims of the present invention. Signals transmitted usingtelecommunications links 110, can be encrypted using such algorithms asBlowfish and other forms of public and private key encryption. Thetelecommunications links 110 can be a dialup connection via a standardmodem 120; a dedicated line connection establishing a local area network(LAN) or wide area network (WAN) 130 running, for example, the Ethernetnetwork protocol; a public Internet connection 140; or wireless orcellular connection 150. Any of the computers and devices 160, 170, 180,190 and 200, depicted in FIG. 1, can be connected using any of the links120, 130, 140 and 150 as depicted in hub 111. Other telecommunicationslinks, such as radio transmission, are known to those of skill in theart.

As depicted in FIG. 1, to establish telecommunications connections withthe central controller 100, a trader or investor can use workstations160 running, for example, UNIX, Windows NT, Linux, or other operatingsystems. In preferred embodiments, the computers used by traders orinvestors include basic input/output capability, can include a harddrive or other mass storage device, a central processor (e.g., anIntel-made Pentium III processor), random-access memory, networkinterface cards, and telecommunications access. A trader or investor canalso use a mobile laptop computer 180, or network computer 190 having,for example, minimal memory and storage capability 190, or personaldigital assistant 200 such as a Palm Pilot. Cellular phones or othernetwork devices may also be used to process and display information fromand communicate with the central controller 100.

FIG. 2 depicts a preferred embodiment of the central controller 100comprising a plurality of software and hardware components. Computerscomprising the central controller 100 are preferably high-endworkstations with resources capable of running business operatingsystems and applications, such as UNIX, Windows NT, SQL Server, andTransaction Server. In a preferred embodiment, these computers arehigh-end personal computers with Intel-made (x86 “instruction set”)CPUs, at least 128 megabytes of RAM, and several gigabytes of hard drivedata storage space. In preferred embodiments, computers depicted in FIG.2 are equipped with JAVA virtual machines, thereby enabling theprocessing of JAVA instructions. Other preferred embodiments of thecentral controller 100 may not require the use of JAVA instruction sets.

In a preferred embodiment of central controller 100 depicted in FIG. 2,a workstation software application server 210, such as the WeblogicServer available from BEA Systems, receives information viatelecommunications links 110 from investors' computers and devices 160,170, 180, 190 and 200. The software application server 210 isresponsible for presenting human-readable user interfaces to investors'computers and devices, for processing requests for services frominvestors' computers and devices, and for routing the requests forservices to other hardware and software components in the centralcontroller 100. The user interfaces that can be available on thesoftware application server 210 include hypertext markup language (HTML)pages, JAVA applets and servlets, JAVA or Active Server pages, or otherforms of network-based graphical user interfaces known to those of skillin the art. For example, investors or traders connected via an Internetconnection for HTML can submit requests to the software applicationserver 210 via the Remote Method Invocation (RMI) and/or the InternetInter-Orb Protocol (IIOP) running on top of the standard TCP/IPprotocol. Other methods are known to those of skill in the art fortransmitting investors' requests and instructions and presenting humanreadable interfaces from the application server 210 to the traders andinvestors. For example, the software application server 210 may hostActive Server Pages and communicate with traders and investors usingDCOM.

In a preferred embodiment, the user interfaces deployed by the softwareapplication server 210 present login, account management, trading,market data, and other input/output information necessary for theoperation of a system for investing in groups of DBAR contingent claimsaccording to the present invention. A preferred embodiment uses the HTMLand JAVA applet/servlet interface. The HTML pages can be supplementedwith embedded applications or “applets” using JAVA based or ActiveXstandards or another suitable application, as known to one of skill inthe art.

In a preferred embodiment, the software application server 210 relies onnetwork-connected services with other computers within the centralcontroller 100. The computers comprising the central controller 100preferably reside on the same local area network (e.g., Ethernet LAN)but can be remotely connected over Internet, dedicated, dialup, or othersimilar connections. In preferred embodiments, networkintercommunication among the computers comprising central controller 100can be implemented using DCOM, CORBA, or TCP/IP or other stack servicesknown to those of skill in the art.

Representative requests for services from the investors' computers tothe software application server 210 include: (1) requests for HTML pages(e.g., navigating and searching a web site); (2) logging onto the systemfor trading DBAR contingent claims; (3) viewing real-time and historicalmarket data and market news; (4) requesting analytical calculations suchas returns, market risk, and credit risk; (5) choosing a group of DBARcontingent claims of interest by navigating HTML pages and activatingJAVA applets; (6) making an investment in one or more defined states ofa group of DBAR contingent claims; and (7) monitoring investments ingroups of DBAR contingent claims.

In a preferred embodiment depicted in FIG. 2, an Object Request Broker(ORB) 230 can be a workstation computer operating specialized softwarefor receiving, aggregating, and marshalling service requests from thesoftware application server 210. For example, the ORB 230 can operate asoftware product called Visibroker, available from Inprise, and relatedsoftware products that provide a number of functions and servicesaccording to the Common Object Request Broker Architecture (CORBA)standard. In a preferred embodiment, one function of the ORB 230 is toprovide what are commonly known in the object-oriented software industryas directory services, which correlate computer code organized intoclass modules, known as “objects,” with names used to access thoseobjects. When an object is accessed in the form of a request by name,the object is instantiated (i.e., caused to run) by the ORB 230. Forexample, in a preferred embodiment, computer code organized into a JAVAclass module for the purpose of computing returns using a canonical DRFis an object named “DRF_Returns,” and the directory services of the ORB230 would be responsible for invoking this object by this name wheneverthe application server 210 issues a request that returns be computed.

In a preferred embodiment, another function of the ORB 230 is tomaintain what is commonly known in the object-oriented software industryas an interface repository, which contains a database of objectinterfaces. The object interfaces contain information regarding whichcode modules perform which functions. For example, in a preferredembodiment, a part of the interface of the object named “DRF_Returns” isa function which fetches the amount currently invested across thedistribution of states for a group of DBAR contingent claims.

In a preferred embodiment, another function of the ORB 230 is to managethe length of runtime for objects which are instantiated by the ORB 230,and to manage other functions such as whether the objects are shared andhow the objects manage memory. For example, in a preferred embodiment,the ORB 230 determines, depending upon the request from the softwareapplication server 210, whether an object which processes market datawill share such data with other objects, such as objects that allocatereturns to investments in defined states.

In a preferred embodiment, another function of the ORB 230 is to providethe ability for objects to communicate asynchronously by responding tomessages or data at varying times and frequencies based upon theactivity of other objects. For example, in a preferred embodiment, anobject that computes returns for a group of DBAR contingent claimsresponds asynchronously in real-time to a new investment andrecalculates returns automatically without a request by the softwareapplication server 210 or any other object. In preferred embodiments,such asynchronous processes are important where computations inreal-time are made in response to other activity in the system, such asa trader making a new investment or the fulfillment of the predeterminedtermination criteria for a group of DBAR contingent claims.

In a preferred embodiment, another function of the ORB 230 is to providefunctions related to what is commonly known in the object-orientedsoftware industry as marshalling. Marshalling in general is the processof obtaining for an object the relevant data it needs to perform itsdesignated function. In preferred embodiments of the present invention,such data includes for example, trader and account information and canitself be manipulated in the form of an object, as is common in thepractice of object-oriented programming. Other functions and servicesmay be provided by the ORB 230, such as the functions and servicesprovided by the Visibroker product, according to the standards andpractices of the object-oriented software industry or as known to thoseof skill in the art.

In a preferred embodiment depicted in FIG. 2, transaction server 240 isa computer running specialized software for performing various tasksincluding: (1) responding to data requests from the ORB 230, e.g., user,account, trade data and market data requests; (2) performing relevantcomputations concerning groups of DBAR contingent claims, such asintra-trading period and end-of-trading-period returns allocations andcredit risk exposures; and (3) updating investor accounts based upon DRFpayouts for groups of DBAR contingent claims and applying debits orcredits for trader margin and positive outstanding investment balances.The transaction server 240 preferably processes all requests from theORB 230 and, for those requests that require stored data (e.g., investorand account information), queries data storage devices 260. In apreferred embodiment depicted in FIG. 2, a market data feed 270 suppliesreal-time and historical market data, market news, and corporate actiondata, for the purpose of ascertaining event outcomes and updatingtrading period returns. The specialized software running on transactionserver 240 preferably incorporates the use of object oriented techniquesand principles available with computer languages such as C++ or Java forimplementing the above-listed tasks.

As depicted in FIG. 2, in a preferred embodiment the data storagedevices 260 can operate relational database software such as Microsoft'sSQL Server or Oracle's 8i Enterprise Server. The types of databaseswithin the data storage devices 260 that can be used to support the DBARcontingent claim and exchange preferably comprise: (1) Trader andAccount databases 261; (2) Market Returns databases 262; (3) Market Datadatabases 263; (4) Event Data databases 264; (5) Risk databases 265; (6)Trade Blotter databases 266; and (7) Contingent Claims Terms andConditions databases 267. The kinds of data preferably stored in eachdatabase are shown in more detail in FIG. 4. In a preferred embodiment,connectivity between data storage devices 260 and transaction server 240is accomplished via TCP/IP and standard Database Connectivity Protocols(DBC) such as the JAVA DBC (JDBC). Other systems and protocols for suchconnectivity are known to those of skill in the art.

In reference to FIG. 2, application server 210 and ORB 230 may beconsidered to form an interface processor, while transaction server 240forms a demand-based transaction processor. Further, the databaseshosted on data storage devices 260 may be considered to form a tradestatus database. Investors, also referred to as traders, communicatingvia telecommunications links 110 from computers and devices 160, 170,180, 190, and 200, may be considered to perform a series of demand-basedinteractions, also referred to as demand-based transactions, with thedemand-based transaction processor. A series of demand-basedtransactions may be used by a trader, for example, to obtain marketdata, to establish a trade, or to close out a trade.

FIG. 3 depicts a preferred embodiment of the implementation of a groupof DBAR contingent claims. As depicted in FIG. 3, an exchange or issuerfirst selects an event of economic significance 300. In the preferredembodiment, the exchange then partitions the possible outcomes for theevent into mutually exclusive and collectively exhaustive states 305,such that one state among the possible states in the partitioneddistribution is guaranteed to occur, and the sum of probabilities of theoccurrence of each partitioned state is unity. Trading can then commencewith the beginning 311 of the first trading period 310. In the preferredembodiment depicted in FIG. 3, a group of DBAR contingent claims hastrading periods 310, 320, 330, and 340, with trading period start date311, 321, 331, 341 respectively, followed by a predetermined timeinterval by each trading period's respective trading end dates 313, 323,333 and 343. The predetermined time interval is preferably of shortduration in order to attain continuity. In the preferred embodiment,during each trading period the transaction server 240 running JAVA codeimplementing the DRF for the group of DBAR contingent claims adjustsreturns immediately in response to changes in the amounts invested ineach of the defined states. Changes in market conditions during atrading period, such as price and volatility changes, as well as changesin investor risk preferences and liquidity conditions in the underlyingmarket, among other factors, will cause amounts invested in each definedstate to change thereby reflecting changes in expectations of tradersover the distribution of states defining the group of DBAR contingentclaims.

In a preferred embodiment, the adjusted returns calculated during atrading period, i.e., intra-trading period returns, are of informationalvalue only—only the returns which are finalized at the end of eachtrading period are used to allocate gains and losses for a trader'sinvestments in a group or portfolio of groups of DBAR contingent claims.In a preferred embodiment, at the end of each trading period, forexample, at trading end dates 313, 323, 333, and 343, finalized returnsare allocated and locked in. The finalized returns are the rates ofreturn to be allocated per unit of amount invested in each defined stateshould that state occur. In a preferred embodiment, each trading periodcan therefore have a different set of finalized returns as marketconditions change, thereby enabling traders to make investments duringlater trading periods which hedge investments from earlier tradingperiods that have since closed.

In another preferred embodiment, not depicted, trading periods overlapso that more than one trading period is open for investment on the sameset of predefined states. For example, an earlier trading period canremain open while a subsequent trading period opens and closes. Otherpermutations of overlapping trading periods are possible and areapparent to one of skill in the art from this specification or practiceof the present invention.

The canonical DRF, as previously described, is a preferred embodiment ofa DRF which takes investment across the distribution of states and eachstate, the transaction fee, and the event outcome and allocates a returnfor each state if it should occur. A canonical DRF of the presentinvention, as previously described, reallocates all amounts invested instates that did not occur to the state that did occur. Each trader thathas made an investment in the state that did occur receives a pro-ratashare of the trades from the non- occurring states in addition to theamount he originally invested in the occurring state, less the exchangefee.

In the preferred embodiment depicted in FIG. 3, at the close of thefinal trading period 343, trading ceases and the outcome for the eventunderlying the contingent claim is determined at close of observationperiod 350. In a preferred embodiment, only the outcome of the eventunderlying the group of contingent claims must be uncertain during thetrading periods while returns are being locked in. In other words, theevent underlying the contingent claims may actually have occurred beforethe end of trading so long as the actual outcome remains unknown, forexample, due to the time lag in measuring or ascertaining the event'soutcome. This could be the case, for instance, with macroeconomicstatistics like consumer price inflation.

In the preferred embodiment depicted in FIG. 2, once the outcome isobserved at time 350, process 360 operates on the finalized returns fromall the trading periods and determines the payouts. In the case of acanonical DRF previously described, the amounts invested in the losinginvestments finance the payouts to the successful investments, less theexchange fee. In a canonical DRF, successful investments are those madeduring a trading period in a state which occurred as determined at time350, and unsuccessful investments are those made in states which did notoccur. Examples 3.1.1-3.1.21 above illustrate various preferredembodiments of a group of DBAR contingent claims using a canonical DRF.In the preferred embodiment depicted in FIG. 3, the results of process360 are made available to traders by posting the results for all tradingperiods on display 370. In a preferred embodiment not depicted, traderaccounts are subsequently updated to reflect these results.

FIG. 4 provides a more detailed depiction of the data storage devices260 of a preferred embodiment of a DBAR contingent claims exchange. In apreferred embodiment, data storage devices 260, on which relationaldatabase software is installed as described above, is a non-volatilehard drive data storage system, which may comprise a single device ormedium, or may be distributed across a plurality of physical devices,such as a cluster of workstation computers operating relational databasesoftware, as described previously and as known in the art. In apreferred embodiment, the relational database software operating on thedata storage devices 260 comprises relational database tables, storedprocedures, and other database entities and objects that are commonlycontained in relational database software packages. In the preferredembodiment depicted in FIG. 4, databases 261-267 each contain suchtables and other relational database entities and objects necessary ordesirable to implement an embodiment of the present invention. FIG. 4identifies the kinds of information that can be stored in such devices.Of course, the kinds of data shown in the drawing are not exhaustive.The storage of other data on the same or additional databases may beuseful depending on the nature of the contingent claim being traded.Moreover, in the preferred embodiment depicted in FIG. 4, certain dataare shown in FIG. 4 as stored in more than one storage device. Invarious other preferred embodiments, such data may be stored in only onesuch device or may be calculated. Other database designs andarchitectures will be apparent to those of skill in the art from thisspecification or practice of the present invention.

In the preferred embodiment depicted in FIG. 4, the Trader and Accountdatabase 261 stores data related to the identification of a DBAR tradersuch as name, password, address, trader identification number, etc. Datarelated to the trader's credit rating can also be stored and updated inresponse to changes in the trader's credit status. Other informationthat can be stored in Trader and Account database 261 includes datarelated to the trader's account, for example, active and inactiveinvestments, the trader's balance, the trader's margin limits,outstanding margin amounts, interest credited on outstanding tradebalances and interest paid on outstanding margin balances, anyrestrictions the trader may have regarding access to his account, andthe trader's profit and loss information regarding active and inactiveinvestments. Information related to multi-state investments to beallocated can also be stored in Trader and Account database 261. Thedata stored in database 261 can be used, for example, to issue accountrelated statements to traders.

In the preferred embodiment depicted in FIG. 4, the Market Returnsdatabase 262 contains information related to returns available atvarious times for active and inactive groups of DBAR contingent claims.In a preferred embodiment, each group of contingent claims in database262 is identified using a unique identifier previously assigned to thatgroup. Returns for each defined state for each group of contingentclaims reflected are stored in database 262. Returns calculated andavailable for display to traders during a given trading period arestored in database 262 for each state and for each claim. At the end ofeach trading period, finalized returns are computed and stored in MarketReturns database 262. Marginal returns, as previously described, canalso be stored in database 262. The data in Market Returns database 262may also include information relevant to a trader's decisions such ascurrent and past intra-period returns, as well as information used todetermine payouts by a DRF for a group of DBAR contingent claims.

In the preferred embodiment depicted in FIG. 4, Market Data database 263stores market data from market data feed 270. In a preferred embodiment,the data in Market Data database 263 include data relevant for the typesof contingent claims which can be traded on a particular exchange. In apreferred embodiment, real-time market data include data such asreal-time prices, yields, index levels, and other similar information.In a preferred embodiment, such real-time data from Market Data database263 are presented to traders to aid in making investment decisions andare used by the DRF to allocate returns for groups of contingent claimswhich depend on such information. Historical data relating to relevantgroups of DBAR contingent claims can also be stored in Market Datadatabase 263. In preferred embodiments, news items related to underlyinggroups of DBAR contingent claims (e.g., comments by the Federal Reserve)are also stored in Market Data database 263 and can be retrieved bytraders.

In the preferred embodiment depicted in FIG. 4, Event Data database 264stores data related to events underlying the groups of DBAR contingentclaims that can be traded on an exchange. In a preferred embodiment,each event is identified by a previously assigned event identificationnumber. Each event has one or more associated group of DBAR contingentclaims based on that event and is so identified with a previouslyassigned contingent claim group identification number. The type of eventcan also be stored in Event database 264, for example, whether the eventis based on a closing price of a security, a corporate earningsannouncement, a previously calculated but yet to be released economicstatistic, etc. The source of data used to determine the outcome of theevent can also be stored in Event database 264. After an event outcomebecomes known, it can also be stored in Event database 264 along withthe defined state of the respective group of contingent claimscorresponding to that outcome.

In the preferred embodiment depicted in FIG. 4, Risk database 265 storesthe data and results and analyses related to the estimation andcalculation of market risk and credit risk. In a preferred embodiment,Risk database 265 correlates the derived results with an accountidentification number. The market and credit risk quantities that can bestored are those related to the calculation of CAR and CCAR, such as thestandard deviation of unit returns for each state, the standarddeviation of dollar returns for each state, the standard deviation ofdollar returns for a given contingent claim, and portfolio CAR.Intermediate estimation and simulation data such as correlation matricesused in VAR-based CAR and CCAR calculations and scenarios used inMCS-based calculations can also be stored in Risk database 265.

In the preferred embodiment depicted in FIG. 4, Trade Blotter database266 contains data related to the investments, both active and inactive,made by traders for all the groups of DBAR contingent claims that can betraded on the particular exchange. Such data may include previouslyassigned trader identification numbers previously assigned investmentidentification numbers, previously assigned account identificationnumbers, previously assigned contingent claim identification numbers,state identification numbers previously assigned corresponding to eachdefined state, the time of each investment, the units of value used tomake each investments (e.g., dollars), the investment amounts, how muchmargin is used to make the investments, and previously assigned tradingperiod identification numbers. In addition, data related to whether aninvestment is a multi-state investment can also be stored. The payoutdistribution which a trader desires to replicate and which the exchangewill implement using a multi-state investment allocation, as describedabove, can also be stored in Trade Blotter database 266.

In the preferred embodiment depicted in FIG. 4, Contingent Claims Termsand Conditions database 267 stores data related to the definition andstructure of each group of DBAR contingent claims. In a preferredembodiment, such data are called “terms and conditions” to indicate thatthey relate to the contractual terms and conditions under which tradersagree to be bound, and roughly correspond to material found inprospectuses in traditional markets. In a preferred embodiment, theterms and conditions provide the fundamental information regarding thenature of the contingent claim to be traded, e.g., the number of tradingperiods, the trading period(s)' start and end times, the type of eventunderlying the contingent claim, how the DRF finances successfulinvestments from unsuccessful investments, the time at which the eventis observed for determining the outcome, other predetermined terminationcriteria, the partition of states in which investments can be made, andthe investment and payout value units (e.g., dollars, numbers of shares,ounces of gold, etc.). In a preferred embodiment, contingent claim andevent identification numbers are assigned and stored in ContingentClaims Terms and Conditions database 267 so that they may be readilyreferred to in other tables of the data storage devices.

FIG. 5 shows a flow diagram depicting illustrative processes used andillustrative decisions made by a trader using a preferred embodiment ofthe present invention. For 5 purposes of illustration in FIG. 5, it isassumed that the trader is making an investment in a DBAR rangederivative (RD) examples of which are disclosed above. In particular, itis assumed for the purposes of illustration that the DBAR RD investmentbeing made is in a contingent claim based upon the closing price of IBMcommon stock on Aug. 3, 1999 (as indicated in the display 501 of FIG.6).

In process 401, depicted in FIG. 5, the trader requests access to theDBAR contingent claim exchange. As previously described in a preferredembodiment, the software application server 210 (depicted in FIG. 2)processes this request and routes it to the ORB 230, which instantiatesan object responsible for the authentication of traders on the exchangeon transaction server 240. The authentication object on transactionserver 240, for example, queries the Trader and Account database 261(depicted in FIG. 4) for the trader's username, password, and otheridentifying information supplied. The authentication object responds byeither allowing or denying access as indicated in process 402 depictedin FIG. 5. If authentication fails in this illustration, process 403prompts the trader to retry a logon or establish valid credentials forlogging onto the system. If the trader has been granted access, thesoftware application server 210 (depicted in FIG. 2) will display to thetrader many user interfaces which may be of interest. For example, in apreferred embodiment, the trader can navigate through a sample of groupsof DBAR contingent claims currently being traded, as represented inprocess 404. The trader may also check current market conditions byrequesting those interfaces in process 404 that contain current marketdata as obtained from market data feed 270 (depicted in FIG. 2) andstored in Market Data database 263 (as depicted in FIG. 4). Process 405of FIG. 5 represents the trader requesting the application server 210for relevant information regarding the trader's account, such as thetrader's current portfolio of trades, trade amounts, current amount ofmargin outstanding, and account balances. In a preferred embodiment,this information is obtained by objects running on transaction server240 (FIG. 2) that query Trader and Account database 261 and TradeBlotter database 266 (FIG. 4).

As depicted in FIG. 5, process 407 represents the selection of a groupof DBAR contingent claims by a trader for the purpose of making aninvestment. The application server 210 (depicted in FIG. 2) can presentuser interfaces to the trader such as the interface shown in FIG. 6 asis known in the art. Process 408 represents the trader requesting dataand analysis which may include calculations as to the effect thetrader's proposed investment would have on the current returns. Thecalculations can be made using the implied “bid” and “offer” demandresponse equations described above. The processes which perform thesedata requests and manipulation of such data are, in a preferredembodiment, objects running on transaction server 240 (as depicted inFIG.2). These objects, for example, obtain data from database 262 (FIG.4) by issuing a query that requests investment amounts across thedistribution of states for a given trading period for a given group ofcontingent claims. With the investment amount data, other objectsrunning on transaction server 240 (FIG. 2) can perform marginal returnscalculations using the DRF of the group of contingent claims asdescribed above. Such processes are objects managed by the ORB 230 (asdepicted in FIG. 2).

Returning to the illustration depicted in FIG. 5, process 411 representsa trader's decision to make an investment for a given amount in one ormore defined states of the group of DBAR contingent claims of interest.In a preferred embodiment, the trader's request to make an investmentidentifies the particular group of claims, the state or states in whichinvestments are to be made, the amount to be invested in the state orstates, and the amount of margin to be used, if any, for theinvestments.

Process 412 responds to any requests to make an investment on margin.The use of margin presents the risk that the exchange may not be able tocollect the entire amount of a losing investment. Therefore, inpreferred embodiments, an analysis is performed to determine the amountof risk to which a current trader is exposed in relation to the amountof margin loans the trader currently has outstanding. In process 413such an analysis is carried out in response to a margin request by thetrader.

The proposed trade or trades under consideration may have the effect ofhedging or reducing the total amount of risk associated with thetrader's active portfolio of investments in groups of DBAR contingentclaims. Accordingly, in a preferred embodiment, the proposed trades andmargin amounts should be included in a CAR analysis of the trader'sportfolio.

In a preferred embodiment, the CAR analysis performed by process 413,depicted in FIG. 5, can be conducted according to the VAR, MCS, or HSmethodologies previously discussed, using data stored in Risk database265 (FIG. 2), such as correlation of state returns, correlation ofunderlying events, etc. In a preferred embodiment, the results of theCAR calculation are also stored in Risk database 265. As depicted inFIG. 5, process 414 determines whether the trader has sufficient equitycapital in his account by comparing the computed CAR value and thetrader's equity in accordance with the exchange's margin rules. Inpreferred embodiments, the exchange requires that all traders maintain alevel of equity capital equal to some portion or multiple of the CARvalue for their portfolios. For example, assuming CAR is computed with a95% statistical confidence as described above, the exchange may requirethat traders have 10 times CAR as equity in their accounts. Such arequirement would mean that traders would suffer drawdowns to equity of10% approximately 5% of the time, which might be regarded as areasonable tradeoff between the benefits of extending margin to tradersto increase liquidity and the risks and costs associated with traderdefault. In addition, in preferred embodiments, the exchange can alsoperform CCAR calculations to determine the amount of credit risk in thegroup of DBAR contingent claims due to each trader. In a preferredembodiment, if a trader does not have adequate equity in his account orthe amount of credit risk posed by the trader is too great, the requestfor margin is denied, as depicted in process 432 (FIG. 5).

As further depicted in FIG. 5, if the trader has requested no margin orthe trader has passed the margin tests applied in process 414, process415 determines whether the investment is one to be made over multiplestates simultaneously in order to replicate a trader's desired payoutdistribution over such states. If the investment is multi-state, process460 requests trader to enter a desired payout distribution. Suchcommunication will comprise, for example, a list of constituent statesand desired payouts in the event that each constituent state occurs. Forexample, for a four-state group of DBAR contingent claims, the tradermight submit the four dimensional vector (10, 0, 5, 2) indicating thatthe trader would like to replicate a payout of 10 value units (e.g.,dollars) should state 1 occur, no payout should state 2 occur, 5 unitsshould state 3 occur, and 2 units should state 4 occur. In a preferredembodiment, this information is stored in Trade Blotter database 266(FIG. 4) where it will be available for the purposes of determining theinvestment amounts to be allocated among the constituent states for thepurposed of replicating the desired payouts. As depicted in FIG. 5, ifthe investment is a multi-state investment, process 417 makes aprovisional allocation of the proposed investment amount to each of theconstituent states.

As further depicted in FIG. 5, the investment details and information(e.g., contingent claim, investment amount, selected state, amount ofmargin, provisional allocation, etc.) are then displayed to the traderfor confirmation by process 416. Process 418 represents the trader'sdecision whether to make the investment as displayed. If the traderdecides against making the investment, it is not executed as representedby process 419. If the trader decides to make the investment and process420 determines that it is not a multi-state investment, the investmentis executed, and the trader's investment amount is recorded in therelevant defined state of the group of DBAR contingent claims accordingto the investment details previously accepted. In a preferredembodiment, the Trade Blotter database 266 (FIG. 4) is then updated byprocess 421 with the new investment information such as the trader ID,trade ID, account identification, the state or states in whichinvestments were made, the investment time, the amount invested, thecontingent claim identification, etc.

In the illustration depicted in FIG. 5, if the trader decides to makethe investment, and process 420 determines that it is a multi-stateinvestment, process 423 allocates the invested amount to the constituentstates comprising the multi-state investment in amounts that generatethe trader's desired payout distribution previously communicated to theexchange in process 460 and stored in Trader Blotter database 266 (FIG.4). For example, in a preferred embodiment, if the desired payouts areidentical payouts no matter which state occurs among the constituentstates, process 423 will update a suspense account entry and allocatethe multi-state trade in proportion to the amounts previously investedin the constituent states. Given the payout distribution previouslystored, the total amount to be invested, and the constituent states inwhich the “new” investment is to be made, then the amount to be investedin each constituent state can be calculated using the matrix formulaprovided in Example 3.1.21, for example. Since these calculations dependon the existing distributions of amounts invested both during and at theend of trading, in a preferred embodiment reallocations are performedwhenever the distribution of amounts invested (and hence returns)change.

As further depicted in FIG. 5, in response to a new investment, Process422 updates the returns for each state to reflect the new distributionof amounts invested across the defined states for the relevant group ofDBAR contingent claims. In particular, process 422 receives the newtrade information from Trade Blotter database 266 as updated by process421, if the investment is not multi-state, or from Trader and Accountdatabase 261 as updated by suspense account process 423, if theinvestment is a multi-state investment. Process 422 involves the ORB 230(FIG. 2) instantiating an object on transaction server 240 forcalculating returns in response to new trades. In this illustration, theobject queries the new trade data from the Trade Blotter database 266 orthe suspense account in Trader and Account database 261 (FIG. 4),computes the new returns using the DRF for the group of contingentclaims, and updates the intra-trading period returns stored in MarketReturns database 262.

As depicted in FIG. 5, if the investment is a multi-state investment asdetermined by process 450, the exchange continues to update the suspenseaccount to reflects the trader's desired payout distribution in responseto subsequent investments entering the exchange. Any updatedintra-trading period returns obtained from process 422 and stored inMarket Returns database 262 are used by process 423 to perform areallocation of multi-state investments to reflect the updated returns.If the trading period has not closed, as determined by process 452, thereallocated amounts obtained from the process 423 are used, along withinformation then simultaneously stored in Trade Blotter database 266(FIG. 4), to perform further intra-trading period update of returns, perprocess 422 shown in FIG. 5. However, if the trading period has closed,as determined in this illustration by process 452, then the multi-statereallocation is performed by process 425 so that the returns for thetrading period can be finalized per process 426.

In a preferred embodiment, the closing of the trading period is animportant point since at that point the DRF object running onTransaction server 240 (FIG. 2) calculates the finalized returns andthen updates Market Returns database 262 with those finalized returns,as represented by process 426 depicted in FIG. 5. The finalized returnsare those which are used to compute payouts once the outcome of theevent and, therefore, the state which occurred are known and all otherpredetermined termination criteria are fulfilled. Even though amulti-state reallocation process 425 is shown in FIG. 5 between process452 and process 426, multi-state reallocation process 425 is not carriedout if the investment is not a multi-state investment.

Continuing with the illustration depicted in FIG. 5, process 427represents the possible existence of subsequent trading periods for thesame event on which the given group of DBAR contingent claims is based.If such periods exist, traders may make investments during them, andeach subsequent trading period would have its own distinct set offinalized returns. For example, the trader in a group of contingentclaims may place a hedging investment in one or more of the subsequenttrading periods in response to changes in returns across the tradingperiods in accordance with the method discussed in Example 3.1.19 above.The ability to place hedging trades in successive trading periods, eachperiod having its own set of finalized returns, allows the trader tolock-in or realize profits and losses in virtually continuous time asreturns change across the trading periods. In a preferred embodiment,the plurality of steps represented by process 427 are performed aspreviously described for the earlier portions of FIG. 5.

As further depicted in FIG. 5, process 428 marks the end of all thetrading periods for a group of contingent claims. In a preferredembodiment, at the end of the last trading period, the Market Returnsdatabase 262 (FIG. 4) contains a set of finalized returns for eachtrading period of the group of contingent claims, and Trade Blotterdatabase 266 contains data on every investment made by every trader onthe illustrative group of DBAR contingent claims.

In FIG. 5, process 429 represents the observation period during whichthe outcome of the event underlying the contingent claim is observed,the occurring state of the DBAR contingent claim determined and anyother predetermined termination criteria are fulfilled. In a preferredembodiment, the event outcome is determined by query of the Market Datadatabase 263 (FIG. 4), which has been kept current by Market Data Feed270. For example, for a group of contingent claims on the event of theclosing price of IBM on Aug. 3, 1999, the Market Data database 263 willcontain the closing price, 119⅜, as obtained from the specified eventdata source in Event Data database 264. The event data source might beBloomberg, in which case an object residing on transaction server 240previously instantiated by ORB 230 will have updated the Market Returnsdatabase 262 with the closing price from Bloomberg. Another similarlyinstantiated object on transaction server 240 will query the MarketReturns database 262 for the event outcome (119⅜), will query theContingent Claims Terms and Conditions database 267 for the purpose ofdetermining the state identification corresponding to the event outcome(e.g., Contingent Claim #1458, state #8) and update the event and stateoutcomes into the Event Data database 264.

As further depicted in FIG. 5, process 430 shows an object instantiatedon transaction server 240 by ORB 230 performing payout calculations inaccordance with the DRF and other terms and conditions as contained inContingent Claims Terms and Conditions database 267 for the given groupof contingent claims. In a preferred embodiment, the object isresponsible for calculating amounts to be paid to successful investmentsand amounts to be collected from unsuccessful investments, i.e.,investments in the occurring and non-occurring states, respectively.

As further depicted in FIG. 5, process 431 shows trader account datastored in Trader and Account database 261 (FIG. 4) being updated by theobject which determines the payouts in process 430. Additionally, inprocess 431 in this illustration and preferred embodiments, outstandingcredit and debit interest corresponding to positive and margin balancesare applied to the relevant accounts in Trader and Account database 261.

FIG. 6 depicts as preferred embodiment of a sample HTML page used bytraders in an exchange for groups of DBAR contingent claims whichillustrates sample display 500 with associated input/output devices,such as display buttons 504-507. As depicted in FIG. 6, descriptive data501 illustrate the basic investment and market information relevant toan investment. In the investment illustrated in FIG. 6, the event is theclosing price of IBM common stock at 4:00 p.m. on Aug. 3, 1999. Asdepicted in FIG. 6, the sample HTML page displays amount invested ineach defined state, and returns available from Market Returns database262 depicted in FIG. 4. In this illustration and in preferredembodiments, returns are calculated on transaction server 240 (FIG. 2)using, for example, a canonical DRF. As also depicted in FIG. 6,real-time market data is displayed in an intraday “tick chart”,represented by display 503, using data obtained from Market Data Feed270, as depicted in FIG. 7, and processed by transaction server 240,depicted in FIG. 2. Market data may also be stored contemporaneously inMarket Data database 263.

In the preferred embodiment depicted in FIG. 6, traders may make aninvestment by selecting Trade button 504. Historical returns and timeseries data, from Market Data database 263 may be viewed by selectingDisplay button 505. Analytical tools for calculating opening orindicative returns or simulating market events are available by requestfrom Software Application Server 210 via ORB 230 and Transaction Server240 (depicted in FIG. 2) by selecting Analyze button 506 in FIG. 6. Asreturns change throughout the trading period, a trader may want todisplay how these returns have changed. As depicted in FIG. 6, theseintraday or intraperiod returns are available from Market Returnsdatabase 262 by selecting Intraday Returns button 507. In addition,marginal intra-period returns, as discussed previously, can be displayedusing the same data in Market Returns database 262 (FIG. 2). In apreferred embodiment, it is also possible for each trader to viewfinalized returns from Market Returns database 262.

In preferred embodiments that are not depicted, display 500 alsoincludes information identifying the group of contingent claims (such asthe claim type and event) available from the Contingent Claims Terms andConditions database 267 or current returns available from Market Returnsdatabase 262 (FIG. 2). In other preferred embodiments, display 500includes means for requesting other services which may be of interest tothe trader, such as the calculation of marginal returns, for example byselecting Intraday Returns button 507, or the viewing of historicaldata, for example by selecting Historical Data button 505.

FIG. 7 depicts a preferred embodiment of the Market Data Feed 270 ofFIG. 2 in greater detail. In a preferred embodiment depicted in FIG. 7,real-time data feed 600 comprises quotes of prices, yields, intradaytick graphs, and relevant market news and example sources. Historicaldata feed 610, which is used to supply market data database 263 withhistorical data, illustrates example sources for market time seriesdata, derived returns calculations from options pricing data, andinsurance claim data. Corporate action data feed 620 depicted in FIG. 7illustrates the types of discrete corporate-related data (e.g., earningsannouncements, credit downgrades) and their example sources which canform the basis for trading in groups of DBAR contingent claims of thepresent invention. In preferred embodiments, functions listed in process630 are implemented on transaction server 240 (FIG. 2) which takesinformation from data feeds 600, 610, and 620 for the purposes ofallocating returns, simulating outcomes, calculating risk, anddetermining event outcomes.

FIG. 8 depicts a preferred embodiment of an illustrative graph ofimplied liquidity effects of investments in a group of DBAR contingentclaims. As discussed above, in preferred embodiments of the presentinvention, liquidity variations within a group of DBAR contingent claimimpose few if any costs on traders since only the finalized or closingreturns for a trading period matter to a trader's return. This contrastswith traditional financial markets, in which local liquidity variationsmay result in execution of trades at prices that do not fairly representfair fundamental value, and may therefore impose permanent costs ontraders.

Liquidity effects from investments in groups of DBAR contingent claims,as illustrated in FIG. 8, include those that occur when an investmentmaterially and permanently affects the distribution of returns acrossthe states. Returns would be materially and perhaps permanently affectedby a trader's investment if, for example, very close to the tradingperiod end time, a trader invested an amount in a state that representeda substantial percentage of aggregate amount previously invested in thatstate. The curves depicted FIG. 8 show in preferred embodiments themaximum effect a trader's investment can have on the distribution ofreturns to the various states in the group of DBAR contingent claims.

As depicted in FIG. 8, the horizontal axis, p, is the amount of thetrader's investment expressed as a percentage of the total amountpreviously invested in the state (the trade could be a multi-stateinvestment, but a single state is assumed in this illustration). Therange of values on the horizontal axis depicted in FIG. 8 has a minimumof 0 (no amount invested) to 10% of the total amount invested in aparticular state. For example, assuming the total amount invested in agiven state is $100 million, the horizontal axis of FIG. 8 ranges from anew investment amount of 0 to $10 million.

The vertical axis of FIG. 8 represents the ratio of the impliedbid-offer spread to the implied probability of the state in which a newinvestment is to be made. In a preferred embodiment, the impliedbid-offer spread is computed as the difference between the implied“offer” demand response, q_(i) ^(O)(ΔT_(i)), and the implied “bid”demand response, q_(i) ^(B)(ΔT_(i)), as defined above. In other words,values along the vertical axis depicted in FIG. 8 are defined by thefollowing ratio:

$\frac{{q_{i}^{O}\left( {\Delta\; T_{i}} \right)} - {q_{i}^{B}\left( {\Delta\; T_{i}} \right)}}{q_{i}}$As displayed in FIG. 8, this ratio is computed using three differentlevels of q_(i), and the three corresponding lines for each level aredrawn over the range of values of p: the ratio is computed assuming alow implied q_(i) (q_(i)=0.091, denoted by the line marked S(p,l)), amiddle-valued q_(i) (q_(i)=0.333, denoted by the line marked S(p,m)),and a high value for q_(i) (q_(i)=0.833 denoted by the line markedS(p,h)), as shown.

If a trader makes an investment in a group of DBAR contingent claims ofthe present invention and there is not enough time remaining in thetrading period for returns to adjust to a fair value, then FIG. 8provides a graphical depiction, in terms of the percentage of theimplied state probability, of the maximum effect a trader's owninvestment can have on the distribution of implied state probabilities.The three separate curves drawn correspond to a high demand and highimplied probability (S(p,h)), medium demand and medium impliedprobability (S(p,m)), and low demand and low implied probability(S(p,l)). As used in this context, the term “demand” means the amountpreviously invested in the particular state.

The graph depicted in FIG. 8 illustrates that the degree to which theamount of a trader's investment affects the existing distribution ofimplied probabilities (and hence returns) varies with the amount ofdemand for the existing state as well as the amount of the trader'sinvestment. If the distribution of implied probabilities is greatlyaffected, this corresponds to a larger implied bid-offer spread, asgraphed on the vertical axis of the graph of FIG. 8. For example, forany given investment amount p, expressed as a percentage of the existingdemand for a particular state, the effect of the new investment amountis largest when existing state demand is smallest (line S(p,l),corresponding to a low demand/low implied probability state). Bycontrast, the effect of the amount of the new investment is smallestwhen the existing state demand is greatest (S(p,h), corresponding to ahigh demand/high implied probability state). FIG. 8 also confirms that,in preferred embodiments, for all levels of existing state demand, theeffect of the amount invested on the existing distribution of impliedprobabilities increases as the amount to be invested increases.

FIG. 8 also illustrates two liquidity-related aspects of groups of DBARcontingent claims of the present invention. First, in contrast to thetraditional markets, in preferred embodiments of the present inventionthe effect of a trader's investment on the existing market can bemathematically determined and calculated and displayed to all traders.Second, as indicated by FIG. 8, the magnitude of such effects are quitereasonable. For example, in preferred embodiments as depicted by FIG. 8,over a wide range of investment amounts ranging up to several percent ofthe existing demand for a given state, the effects on the market of suchinvestments amounts are relatively small. If the market has time toadjust after such investments are added to demand for a state, theeffects on the market will be only transitory and there may be no effecton the implied distribution of probabilities owing to the trader'sinvestment. FIG. 8 illustrates a “worst case” scenario by implicitlyassuming that the market does not adjust after the investment is addedto the demand for the state.

FIGS. 9 a to 9 c illustrate, for a preferred embodiment of a group ofDBAR contingent claims, the trader and credit relationships and howcredit risk can be quantified, for example in process 413 of FIG. 5.FIG. 9 a depicts a counterparty relationship for a traditional swaptransaction, in which two counterparties have previously entered into a10-year swap which pays a semi-annual fixed swap rate of 7.50%. Thereceiving counterparty 701 of the swap transaction receives the fixedrate and pays a floating rate, while the paying counterparty 702 paysthe fixed rate and receives the floating rate. Assuming a $100 millionswap trade and a current market fixed swap rate of 7.40%, based uponwell-known swap valuation principles implemented in software packagessuch as are available from Sungard Data Systems, the receivingcounterparty 701 would receive a profit of $700,000 while the payingswap counterparty 702 would have a loss of $700,000. The receiving swapcounterparty 701 therefore has a credit risk exposure to the paying swapcounterparty 702 as a function of $700,000, because the arrangementdepends on the paying swap party 702 meeting its obligation.

FIG. 9 b depicts illustrative trader relationships in which a preferredembodiment of a group of the DBAR contingent claims and exchangeeffects, as a practical matter, relationships among all the traders. Asdepicted in FIG. 9 b, traders C1, C2, C3, C4, and C5 each have investedin one or more states of a group of DBAR contingent claims, with definedstates S1 to S8 respectively corresponding to ranges of possibleoutcomes for the 10 year swap rate, one year forward. In thisillustration, each of the traders has a credit risk exposure to all theothers in relation to the amount of each trader's investment, how muchof each investment is on margin, the probability of success of eachinvestment at any point in time, the credit quality of each trader, andthe correlation between and among the credit ratings of the traders.This information is readily available in preferred embodiments of DBARcontingent claim exchanges, for example in Trader and Account database261 depicted in FIG. 2, and can be displayed to traders in a formsimilar to tabulation 720 shown in FIG. 9 c, where the amount ofinvestment margin in each state is displayed for each trader, juxtaposedwith that trader's credit rating. For example, as depicted in FIG. 9 c,trader C1 who has a AAA credit rating has invested $50,000 on margin instate 7 and $100,000 on margin in state 8. In a preferred embodiment,the amount of credit risk borne by each trader can be ascertained, forexample using data from Market Data database 263 on the probability ofchanges in credit ratings (including probability of default), amountsrecoverable in case of default, correlations of credit rating changesamong the traders and the information displayed in tabulation 720.

To illustrate such determinations in the context of a group of DBARcontingent claims depicted in FIG. 9 c, the following assumptions aremade: (i) all the traders C1, C2, C3, C4 and C5 investing in the groupof contingent claims have a credit rating correlation of 0.9; (ii) theprobabilities of total default for the traders C1 to C5 are (0.001,0.003, 0.007, 0.01, 0.02) respectively; (iii) the implied probabilitiesof states S1 to S8 (depicted in FIG. 9 c) are(0.075,0.05,0.1,0.25,0.2,0.15,0.075,0.1), respectively. A calculationcan be made with these assumptions which approximates the total creditrisk for all of the traders in the group of the DBAR contingent claimsof FIG. 9 c, following Steps (i)-(vi) previously described for using VARmethodology to determine Credit-Capital-at-Risk.

Step (i) involves obtaining for each trader the amount of margin used tomake each trade. For this illustration, these data are assumed and aredisplayed in FIG. 9 c, and in a preferred embodiment, are available fromTrader and Account database 261 and Trade Blotter database 266.

Step (ii) involves obtaining data related to the probability of defaultand the percentage of outstanding margin loans that are recoverable inthe event of default. In preferred embodiments, this information isavailable from such sources as the JP Morgan CreditMetrics database. Forthis illustration a recovery percentage of zero is assumed for eachtrader, so that if a trader defaults, no amount of the margin loan isrecoverable.

Step (iii) involves scaling the standard deviation of returns (in unitsof the amounts invested) by the percentage of margin used for eachinvestment, the probability of default for each trader, and thepercentage not recoverable in the event of default. For thisillustration, these steps involve computing the standard deviations ofunit returns for each state, multiplying by the margin percentage ineach state, and then multiplying this result by the probability ofdefault for each trader. In this illustration, using the assumed impliedprobabilities for states 1 through 8, the standard deviations of unitreturns are: (3.5118, 4.359,3,1.732,2,2.3805,3.5118,3). In thisillustration these unit returns are then scaled by multiplying each by(a) the amount of investment on margin in each state for each trader,and (b) the probability of default for each trader, yielding thefollowing table:

S1 S2 S3 S4 55 S6 57 S8 C1, 175.59 300 AAA C2, 285.66 263.385 AA C3,1400 999.81 AA C4, 2598 2000 A+ C5, 7023.6 4359 4800 A

Step (iv) involves using the scaled amounts, as shown in the above tableand a correlation matrix C_(s) containing a correlation of returnsbetween each pair of defined states, in order to compute aCredit-Capital-At-Risk. As previously discussed, this Step (iv) isperformed by first arranging the scaled amounts for each trader for eachstate into a vector U as previously defined, which has dimension equalto the number of states (e.g., 8 in this example). For each trader, thecorrelation matrix C_(s) is pre-multiplied by the transpose of U andpost-multiplied by U. The square root of the result is acorrelation-adjusted CCAR value for each trader, which represents theamount of credit risk contributed by each trader. To perform thesecalculations in this illustration, the matrix C_(s) having 8 rows and 8columns and 1's along the diagonal is constructed using the methodspreviously described:

$C_{s} = \begin{matrix}1 & {- {.065}} & {- {.095}} & {- {.164}} & {- {.142}} & {- {.12}} & {- {.081}} & {- {.095}} \\{- {.065}} & 1 & {- {.076}} & {- {.132}} & {- {.115}} & {- {.096}} & {- {.065}} & {- {.076}} \\{- {.095}} & {- {.076}} & 1 & {- {.192}} & {- {.167}} & {- {.14}} & {- {.095}} & {- {.111}} \\{- {.164}} & {- {.132}} & {- {.192}} & 1 & {- {.289}} & {- {.243}} & {- {.164}} & {- {.192}} \\{- {.142}} & {- {.115}} & {- {.167}} & {- {.289}} & 1 & {- {.21}} & {- {.142}} & {- {.167}} \\{- {.12}} & {- {.096}} & {- {.14}} & {- {.243}} & {- {.21}} & 1 & {- {.12}} & {- {.14}} \\{- {.081}} & {- {.065}} & {- {.095}} & {- {.164}} & {- {.142}} & {- {.12}} & 1 & {- {.095}} \\{- {.095}} & {- {.076}} & {- {.111}} & {- {.192}} & {- {.167}} & {- {.14}} & {- {.095}} & 1\end{matrix}$The vectors U₁, U₂, U₃, U₄, and U₅ for each of the 5 traders in thisillustration, respectively, are as follows:

$U_{1} = {{\begin{matrix}0 \\0 \\0 \\0 \\0 \\0 \\175.59 \\300\end{matrix}U_{2}} = {{\begin{matrix}0 \\0 \\0 \\0 \\0 \\285.66 \\263.385 \\0\end{matrix}U_{3}} = {{\begin{matrix}0 \\0 \\0 \\0 \\1400 \\999.81 \\0 \\0\end{matrix}U_{4}} = {{\begin{matrix}0 \\0 \\0 \\2598 \\2000 \\0 \\0 \\0\end{matrix}U_{5}} = \begin{matrix}7023.6 \\4359 \\4800 \\0 \\0 \\0 \\0 \\0\end{matrix}}}}}$Continuing with the methodology of Step (iv) for this illustration, fivematrix computations are performed as follows:CCAR_(i) =√{square root over (U_(i) ^(T) *C _(s) *U _(i))}for i=1 . . . 5. The left hand side of the above equation is the creditcapital at risk corresponding to each of the five traders.

Pursuant to Step (v) of the CCAR methodology as applied to this example,the five CCAR values are arranged into a column vector of dimensionfive, as follows:

$w_{CCAR} = \begin{matrix}332.9 \\\begin{matrix}364.58 \\1540.04 \\2783.22 \\8820.77\end{matrix}\end{matrix}$

Continuing with this step, a correlation matrix (CCAR) with a number ofrows and columns equal to the number of traders is constructed whichcontains the statistical correlation of changes in credit ratingsbetween every pair of traders on the off-diagonals and 1's along thediagonal. For the present example, the final Step (vi) involves thepre-multiplication of CCAR by the transpose of w_(CCAR) and the postmultiplication of C_(CCAR) by w_(CCAR) and taking the square root ofthat product, as follows:CCAR_(TOTAL) =√{square root over (w_(CCAR) ^(T) *C _(CCAR) *w _(CCAR))}In this illustration, the result of this calculation is:

${CCAR}_{TOTAL} = \sqrt{\begin{matrix}33.29 & 364.58 & 1540.04 & 2783.22 & {{8820.77*\begin{matrix}1 & {.9} & {.9} & {.9} & {.9} & 332.9 \\{.9} & 1 & {.9} & {.9} & {.9} & 364.58 \\{.9} & {.9} & 1 & {.9} & {.9} & {*1540.04} \\{.9} & {.9} & {.9} & 1 & {.9} & 2783.22 \\{.9} & {.9} & {.9} & {.9} & 1 & 8820.77\end{matrix}} = 13462.74}\end{matrix}}$

In other words, in this illustration, the margin total and distributionshowing in FIG. 9 c has a single standard deviationCredit-Capital-At-Risk of $13,462.74. As described previously in thediscussion of Credit-Capital-At-Risk using VAR methodology, this amountmay be multiplied by a number derived using methods known to those ofskill in the art in order to obtain a predetermined percentile of creditloss which a trader could believe would not be exceeded with apredetermined level of statistical confidence. For example, in thisillustration, if a trader is interested in knowing, with a 95%statistical confidence, what loss amount would not be exceeded, thesingle deviation Credit-Capital-At-Risk figure of $13,462.74 would bemultiplied by 1.645, to yield a figure of $22,146.21.

A trader may also be interested in knowing how much credit risk theother traders represent among themselves. In a preferred embodiment, thepreceding steps (i)-(vi) can be performed excluding one or more of thetraders. For example, in this illustration, the most risky trader,measured by the amount of CCAR associated with it, is trader C5. Theamount of credit risk due to C1 through C4 can be determined byperforming the matrix calculation of Step (v) above, by entering 0 forthe CCAR amount of trader C5. This yields, for example, a CCAR fortraders C1 through C4 of $4,870.65.

FIG. 10 depicts a preferred embodiment of a feedback process forimproving of a system or exchange for implementing the presentinvention. As depicted in FIG. 10, in a preferred embodiment, closingand intraperiod returns from Market Returns database 262 and market datafrom Market Data database 263 (depicted in FIG. 2) are used by process910 for the purpose of evaluating the efficiency and fairness of theDBAR exchange. One preferred measure of efficiency is whether adistribution of actual outcomes corresponds to the distribution asreflected in the finalized returns. Distribution testing routines, suchas Kolmogorov-Smimoff tests, preferably are performed in process 910 todetermine whether the distributions implied by trading activity in theform of returns across the defined states for a group of DBAR contingentclaims are significantly different from the actual distributions ofoutcomes for the underlying events, experienced over time. Additionally,in preferred embodiments, marginal returns are also analyzed in process910 in order to determine whether traders who make investments late inthe trading period earn returns statistically different from othertraders. These “late traders,” for example, might be capturinginformational advantages not available to early traders. In response tofindings from analyses in process 910, a system according to the presentinvention for trading and investing in groups of the DBAR contingentclaims can be modified to improve its efficiency and fairness. Forexample, if “late traders” earn unusually large profits, it could meanthat such a system is being unfairly manipulated, perhaps in conjunctionwith trading in traditional security markets. Process 920 depicted inFIG. 10 represents a preferred embodiment of a counter-measure whichrandomizes the exact time at which a trading period ends for thepurposes of preventing manipulation of closing returns. For example, ina preferred embodiment, an exchange announces a trading closing end timefalling randomly between 2:00 p.m and 4:00 p.m on a given date.

As depicted in FIG. 10, process 923 is a preferred embodiment of anotherprocess to reduce risk of market manipulation. Process 923 representsthe step of changing the observation period or time for the outcome. Forexample, rather than observing the outcome at a discrete time, theexchange may specify that a range of times for observation will used,perhaps spanning many hours, day, or weeks (or any arbitrary timeframe), and then using the average of the observed outcomes to determinethe occurrence of a state.

As further depicted in FIG. 10, in response to process 910, steps couldbe taken in process 924 to modify DRFs in order, for example, toencourage traders to invest earlier in a trading period. For example, aDRF could be modified to provide somewhat increased returns to these“early” traders and proportionately decreased returns to “late” traders.Such incentives, and others apparent to those skilled in the art, couldbe reflected in more sophisticated DRFs.

In a preferred embodiment depicted in FIG. 10, process 921 represents,responsive to process 910, steps to change the assumptions under whichopening returns are computed for the purpose of providing better openingreturns at the opening of the trading period. For example, the resultsof process 910 might indicate that traders have excessively traded theextremes of a distribution in relation to actual outcomes. There isnothing inherently problematic about this, since trader expectations forfuture possible outcomes might reflect risk preferences that cannot beextracted or analyzed with actual data. However, as apparent to one ofskill in the art, it is possible to adjust the initial returns toprovide better estimates of the future distribution of states, by, forexample, adjusting the skew, kurtosis, or other statistical moments ofthe distribution.

As depicted in FIG. 10, process 922 illustrates changing entirely thestructure of one or more groups of DBAR contingent claims. Such acountermeasure can be used on an ad hoc basis in response to graveinefficiencies or unfair market manipulation. For example, process 922can include changes in the number of trading periods, the timing oftrading periods, the duration of a group of DBAR contingent claims, thenumber of and nature of the defined state partitions in order to achievebetter liquidity and less unfair market manipulation for groups of DBARcontingent claims of the present invention.

The foregoing detailed description of the figures, and the figuresthemselves, are designed to provide and explain specific illustrationsand examples of preferred embodiments of methods and systems of thepresent invention. The purpose is to facilitate increased understandingand appreciation of the present invention. The detailed description andfigures are not meant to limit either the scope of the invention, itsembodiments, or the ways in which it may be implemented or practiced. Tothe contrary, additional embodiments and their equivalents within thescope of this invention will be apparent to those of skill in the artfrom reviewing this specification or practicing this invention.

7 ADVANTAGES OF PREFERRED EMBODIMENTS

This specification sets forth principles, methods, and systems thatprovide trading and investment in groups of DBAR contingent claims, andthe establishment and operation of markets and exchanges for suchclaims. Advantages of the present invention as it applies to the tradingand investment in derivatives and other contingent claims include:

-   -   (1) Increased liquidity: Groups of DBAR contingent claims and        exchanges for investing in them according to the present        invention offer increased liquidity for the following reasons:        -   (i) Reduced dynamic hedging by market makers. In preferred            embodiments, an exchange or market maker for contingent            claims does not need to hedge in the market. In such            embodiments, all that is required for a well-functioning            contingent claims market is a set of observable underlying            real-world events reflecting sources of financial or            economic risk. For example, the quantity of any given            financial product available at any given price can be            irrelevant in a system of the present invention.        -   (ii) Reduced order crossing. Traditional and electronic            exchanges typically employ sophisticated algorithms for            market and limit order book bid/offer crossing. In preferred            embodiments of the present invention, there are no bids and            offers to cross. A trader who desires to “unwind” an            investment will instead make a complementary investment,            thereby hedging his exposure.        -   (iii) No permanent liquidity charge: In the DBAR market,            only the final returns are used to compute payouts.            Liquidity variations and the vagaries of execution in the            traditional markets do not, in preferred embodiments, impose            a permanent tax or toll as they typically do in traditional            markets. In any event, in preferred embodiments of the            present invention, liquidity effects of amounts invested in            groups of DBAR claims are readily calculable and available            to all traders. Such information is not readily available in            traditional markets.    -   (2) Reduced credit risk: In preferred embodiments of the present        invention, the exchange or dealer has greatly increased        assurance of recovering its transaction fee. It therefore has        reduced exposure to market risk. In preferred embodiments, the        primary function of the exchange is to redistribute returns to        successful investments from losses incurred by unsuccessful        investments. By implication, traders who use systems of the        present invention can enjoy limited liability, even for short        positions, and a diversification of counterparty credit risk.    -   (3) Increased Scalability: The pricing methods in preferred        embodiments of systems and methods of the present invention for        investing in groups of DBAR contingent claims are not tied to        the physical quantity of underlying financial products available        for hedging. In preferred embodiments an exchange therefore can        accommodate a very large community of users at lower marginal        costs.    -   (4) Improved Information Aggregation: Markets and exchanges        according to the present invention provide mechanisms for        efficient aggregation of information related to investor demand,        implied probabilities of various outcomes, and price.    -   (5) Increased Price Transparency: Preferred embodiments of        systems and methods of the present invention for investing in        groups of DBAR contingent claims determine returns as functions        of amounts invested. By contrast, prices in traditional        derivatives markets are customarily available for fixed        quantities only and are typically determined by complex        interactions of supply/demand and overall liquidity conditions.        For example, in a preferred embodiment of a canonical DRF for a        group of DBAR contingent claims of the present invention,        returns for a particular defined state are allocated based on a        function of the ratio of the total amount invested across the        distribution of states to the amount on the particular state.    -   (6) Reduced settlement or clearing costs: In preferred        embodiments of systems and methods for investing in groups of        DBAR contingent claims, an exchange need not, and typically will        not, have a need to transact in the underlying physical        financial products on which a group of DBAR contingent claims        may be based.

Securities and derivatives in those products need not be transferred,pledged, or otherwise assigned for value by the exchange, so that, inpreferred embodiments, it does not need the infrastructure which istypically required for these back office activities.

-   -   (7) Reduced hedging costs: In traditional derivatives markets,        market makers continually adjust their portfolio of risk        exposures in order to mitigate risks of bankruptcy and to        maximize expected profit. Portfolio adjustments, or dynamic        hedges, however, are usually very costly. In preferred        embodiments of systems and methods for investing in groups of        DBAR contingent claims, unsuccessful investments hedge the        successful investments. As a consequence, in such preferred        embodiments, the need for an exchange or market maker to hedge        is greatly reduced, if not eliminated.    -   (8) Reduced model risk: In traditional markets, derivatives        dealers often add “model insurance” to the prices they quote to        customers to protect against unhedgable deviations from prices        otherwise indicated by valuation models. In the present        invention, the price of an investment in a defined state derives        directly from the expectations of other traders as to the        expected distribution of market returns. As a result, in such        embodiments, sophisticated derivative valuation models are not        essential. Transaction costs are thereby lowered due to the        increased price transparency and tractability offered by the        systems and methods of the present invention.    -   (9) Reduced event risk: In preferred embodiments of systems and        methods of the present invention for investing in groups of DBAR        contingent claims, trader expectations are solicited over an        entire distribution of future event outcomes.

In such embodiments, expectations of market crashes, for example, aredirectly observable from indicated returns, which transparently revealtrader expectations for an entire distributions of future eventoutcomes. Additionally, in such embodiments, a market maker or exchangebears greatly reduced market crash or “gap” risk, and the costs ofderivatives need not reflect an insurance premium for discontinuousmarket events.

-   -   (10) Generation of Valuable Data: Traditional financial product        exchanges usually attach a proprietary interest in the real-time        and historical data that is generated as a by-product from        trading activity and market making. These data include, for        example, price and volume quotations at the bid and offer side        of the market. In traditional markets, price is a “sufficient        statistic” for market participants and this is the information        that is most desired by data subscribers. In preferred        embodiments of systems and methods of the present invention for        investing in groups of DBAR contingent claims, the scope of data        generation may be greatly expanded to include investor        expectations of the entire distribution of possible outcomes for        respective future events on which a group of DBAR contingent        claims can be based. This type of information (e.g., did the        distribution at time t reflect traders' expectations of a market        crash which occurred at time t+1?) can be used to improve market        operation. Currently, this type of distributional information        can be derived only with great difficulty by collecting panels        of option price data at different strike prices for a given        financial product, using the methods originated in 1978 by the        economists Litzenberger and Breeden and other similar methods        known to someone of skill in the art. Investors and others must        then perform difficult calculations on these data to extract        underlying distributions. In preferred embodiments of the        present invention, such distributions are directly available.    -   (11) Expanded Market For Contingent Claims: Another advantage of        the present invention is that it enables a well functioning        market for contingent claims. Such a market enables traders to        hedge directly against events that are not readily hedgeable or        insurable in traditional markets, such as changes in mortgage        payment indices, changes in real estate valuation indices, and        corporate earnings announcements. A contingent claims market        operating according to the systems and methods of the present        invention can in principle cover all events of economic        significance for which there exists a demand for insurance or        hedging.    -   (12) Price Discovery: Another advantage of systems and methods        of the present invention for investing in groups of DBAR        contingent claims is the provision, in preferred embodiments, of        a returns adjustment mechanism (“price discovery”). In        traditional capital markets, a trader who takes a large position        in relation to overall liquidity often creates the risk to the        market that price discovery will break down in the event of a        shock or liquidity crisis. For example, during the fall of 1998,        Long Term Capital Management (LTCM) was unable to liquidate its        inordinately large positions in response to an external shock to        the credit market, i.e., the pending default of Russia on some        of its debt obligations. This risk to the system was        externalized to not only the creditors of LTCM, but also to        others in the credit markets for whom liquid markets        disappeared. By contrast, in a preferred embodiment of a group        of DBAR contingent claims according to the present invention,        LTCM's own trades in a group of DBAR contingent claims would        have lowered the returns to the states invested in dramatically,        thereby reducing the incentive to make further large, and        possibly destabilizing, investments in those same states.        Furthermore, an exchange for a group of DBAR contingent claims        according to the present invention could still have operated,        albeit at frequently adjusted returns, even during the most        acute phases of the 1998 Russian bond crisis. For example, had a        market in a DBAR range derivative existed which elicited trader        expectations on the distribution of spreads between high-grade        United States Treasury securities and lower-grade debt        instruments, LTCM could have “hedged” its own speculative        positions in the lower-grade instruments by making investment in        the DBAR range derivatives in which it would profit as credit        spreads widened. Of course, its positions by necessity would        have been sizable thereby driving the returns on its position        dramatically lower (i.e., effectively liquidating its existing        position at less favorable prices). Nevertheless, an exchange        according to preferred embodiments of the present invention        could have provided increased liquidity compared to that of the        traditional markets.    -   (13) Improved Offers of Liquidity to the Market: As explained        above, in preferred embodiments of groups of DBAR contingent        claims according to the present invention, once an investment        has been made it can be offset by making an investment in        proportion to the prevailing traded amounts invested in the        complement states and the original invested state. By not        allowing trades to be removed or cancelled outright, preferred        embodiments promote two advantages:        -   1. reducing strategic behavior (“returns-jiggling”)        -   2. increasing liquidity to the market    -   In other words, preferred embodiments of the present invention        reduce the ability of traders to make and withdraw large        investments merely to create false-signals to other participants        in the hopes of creating last-minute changes in closing returns.        Moreover, in preferred embodiments, the liquidity of the market        over the entire distribution of states is information readily        available to traders and such liquidity, in preferred        embodiments, may not be withdrawn during the trading periods.        Such preferred embodiments of the present invention thus provide        essentially binding commitments of liquidity to the market        guaranteed not to disappear.    -   (14) Increased Liquidity Incentives: In preferred embodiments of        the systems and methods of the present invention for trading or        investing in groups of DBAR contingent claims, incentives are        created to provide liquidity over the distribution of states        where it is needed most. On average, in preferred embodiments,        the implied probabilities resulting from invested amounts in        each defined state should be related to the actual probabilities        of the states, so liquidity should be provided in proportion to        the actual probabilities of each state across the distribution.        The traditional markets do not have such ready        self-equilibrating liquidity mechanisms—e.g., far        out-of-the-money options might have no liquidity or might be        excessively traded. In any event, traditional markets do not        generally provide the strong (analytical) relationship between        liquidity, prices, and probabilities so readily available in        trading in groups of DBAR contingent claims according to the        present invention.    -   (15) Improved Self-Consistency: Traditional markets customarily        have “no-arbitrage” relationships such as put-call parity for        options and interest-rate parity for interest rates and        currencies. These relationships typically must (and do) hold to        prevent risk-less arbitrage and to provide consistency checks or        benchmarks for no-arbitrage pricing. In preferred embodiments of        systems and methods of the present invention for trading or        investing in groups of DBAR contingent claims, in addition to        such “no-arbitrage” relationships, the sum of the implied        probabilities over the distribution of defined states is known        to all traders to equal unity. Using the notation developed        above, the following relations hold for a group of DBAR        contingent claims using a canonical DRF:

$r_{i} = {\frac{\left( {1 - f} \right)*{\sum\limits_{i}\; T_{i}}}{T_{i}} - 1}$$q_{i} = {\frac{1 - f}{r_{i} + 1} = \frac{T_{i}}{\sum\limits_{i}\; T_{i}}}$${\sum\limits_{i}\; q_{i}} = 1$

-   -   In other words, in a preferred embodiment, the sum across a        simple function of all implied probabilities is equal to the sum        of the amount traded for each defined state divided by the total        amount traded. In such an embodiment, this sum equals 1. This        internal consistency check has no salient equivalent in the        traditional markets where complex calculations are typically        required to be performed on illiquid options price data in order        to recover the implied probability distributions.    -   (16) Facilitated Marginal Returns Calculations: In preferred        embodiments of systems and methods of the present invention for        trading and investing in groups of DBAR contingent claims,        marginal returns may also be calculated readily. Marginal        returns r^(m) are those that prevail in any sub-period of a        trading period, and can be calculated as follows:

$r_{i,{t - 1},t}^{m} = \frac{{r_{i,t}*T_{i,t}} - {r_{i,{t - 1}}*T_{i,{t - 1}}}}{T_{i,t} - T_{i,{t - 1}}}$where the left hand side is the marginal returns for state i betweentimes t−1 and t:r_(i,t) and r_(i,t−1) are the unit returns for state iat times t, and t−1, and T_(i,t) and T_(i,t−1) are the amounts investedin state i at times t and t−1, respectively.

-   -   In preferred embodiments, the marginal returns can be displayed        to provide important information to traders and others as to the        adjustment throughout a trading period. In systems and methods        of the present invention, marginal returns may be more variable        (depending on the size of the time increment among other        factors) than the returns which apply to the entire trading        period.    -   (17) Reduced Influence By Market Makers: In preferred        embodiments of the systems and methods of the present invention,        because returns are driven by demand, the role of the supply        side market maker is reduced if not eliminated. A typical market        maker in the traditional markets (such as an NYSE specialist or        a swaps book-runner) typically has privileged access to        information (e.g., the limit order book) and potential conflicts        of interest deriving from dual roles as principal (i.e.,        proprietary trader) and market maker. In preferred embodiments        of the present invention, all traders have greater information        (e.g., investment amounts across entire distribution of states)        and there is no supply-side conflict of interest.    -   (18) Increased Ability to Generate and Replicate Arbitrary        Payout Distributions: In preferred embodiments of the systems        and methods of the present invention for investing and trading        in groups of DBAR contingent claims, traders may generate their        own desired distributions of payouts, i.e., payouts can be        customized very readily by varying amounts invested across the        distribution of defined states. This is significant since groups        of DBAR contingent claims can be used to readily replicate        payout distributions with which traders are familiar from the        traditional markets, such as long stock positions, long and        short futures positions, long options straddle positions, etc.        Importantly, as discussed above, in preferred embodiments of the        present invention, the payout distributions corresponding to        such positions can be effectively replicated with minimal        expense and difficulty by having a DBAR contingent claim        exchange perform multi-state allocations.

Preferred embodiments of the invention have been described in detailabove, various changes thereto and equivalents thereof will be readilyapparent to one of ordinary skill in the art and are encompassed withinthe scope of this invention and the appended claim. For example, manytypes of demand reallocation functions (DRFs) can be employed to financegains to successful investments with losses from unsuccessfulinvestments, thereby achieving different risk and return profiles totraders. Additionally, this disclosure has primarily discussed methodsand systems for groups and portfolios of DBAR contingent claims, andmarkets and exchanges for those groups. The methods and systems of thepresent invention can readily be adapted by financial intermediaries foruse within the traditional capital and insurance markets. For example, agroup of DBAR contingent claims can be embedded within a traditionalsecurity, such as a bond for a given corporate issuer, and underwrittenand issued by an underwriter as previously discussed. It is alsointended that such embodiments and their equivalents are encompassed bythe present invention and the appended claims.

The present invention has been described above in the context of tradingderivative securities, specifically the implementation of an electronicderivatives exchange which facilitates the efficient trading of (i)financial-related contingent claims such as stocks, bonds, andderivatives thereon, (ii) non-financial related contingent claims suchas energy, commodity, and weather derivatives, and (iii) traditionalinsurance and reinsurance contracts such as market loss warranties forproperty-casualty catastrophe risk. The present invention is not limitedto these contexts, however, and can be readily adapted to any contingentclaim relating to events which are currently uninsurable or unhedgable,such as corporate earnings announcements, future semiconductor demand,and changes in technology. These and all other such modifications areintended to fall within the scope of the present invention.

8 TECHNICAL APPENDIX Financial Products Having Demand-Based, AdjustableReturns, and Trading Exchange Therefore

This technical appendix provides the mathematical foundation underlyingthe computer code listing of Table I: Illustrative Visual Basic ComputerCode for Solving CDRF 2. The aforementioned computer code listingimplements a procedure for solving the Canonical Demand ReallocationFunction (CDRF 2) by preferred means which one of ordinary skill in theart will recognize are based upon the application of a mathematicalmethod known as fixed point iteration.

As previously indicated in the specification, the simultaneous systemembodied by CDRF 2 does not provide an explicit solution and typicallywould require the use of numerical methods to solve the simultaneousquadratic equations included in the system. In general, such systemswould typically be solved by what are commonly known as “grid search”routines such as the Newton-Raphson method, in which an initial solutionor guess at a solution is improved by extracting information from thenumerical derivatives of the functions embodied in the simultaneoussystem.

One of the important advantages of the demand-based trading methods ofthe present invention is the careful construction of CDRF 2 which allowsfor the application of fixed point iteration as a means for providing anumerical solution of CDRF 2. Fixed point iteration means are generallymore reliable and computationally less burdensome than grid searchroutines, as the computer code listing in Table I illustrates.

A. Fixed Point Iteration

The solution to CDRF 2 requires finding a fixed point to a system ofequations. Fixed points represent solutions since they convey theconcept of a system at “rest” or equilibrium, i.e., a fixed point of asystem of functions or transformations denoted g(a) exists ifα=g(α)

Mathematically, the function g(a) can be said to be a map on the realline over the domain of a. The map, g(x), generates a new point, say, y,on the real line. If x=y, then x is called a fixed point of the functiong(a). In terms of numerical solution techniques, if g(a) is a non-linearsystem of equations and if x is a fixed point of g(a), then a is alsothe zero of the function. If no fixed points such as x exist for thefunction g(a), then grid search type routines can be used to solve thesystem (e.g., the Newton-Raphson Method, the Secant Method, etc.). If afixed point exists, however, its existence can be exploited in solvingfor the zero of a simultaneous non-linear system, as follows.

Choose an initial starting point, x₀, which is believed to be somewherein the neighborhood of the fixed point of the function g(a). Then,assuming there does exist a fixed point of the function g(a), employ thefollowing simple iterative scheme:x _(i+1) =g(x _(i)), where x₀ is chosen as starting pointwhere i=0,1,2, . . . n. The iteration can be continued until a desiredprecision level,ε, is achieved, i.e.,x _(n) =g(x _(n−1)), until|g(x _(n−1))−x _(n)|<εThe question whether fixed point iteration will converge, of course,depends crucially on the value of the first derivative of the functiong(x) in the neighborhood of the fixed point as shown in the followingfigure:As previously indicated, an advantage of the present invention is theconstruction of CDRF 2 in such a way so that it may be represented interms of a multivariate function, g(x), which is continuous and has aderivative whose value is between 0 and 1, as shown below.

B. Fixed Point Iteration as Applied to CDRF 2

This section will demonstrate that (1) the system of equations embodiedin CDRF 2 possesses a fixed point solution and (2) that this fixed pointsolution can be located using the method of fixed point iterationdescribed in Section A, above.

The well known fixed point theorem provides that, if g:[a,b]→[a,b] iscontinuous on [a,b] and differentiable on (a,b) and there is a constantk<1 such that for all x in (a,b),|g′(x)|≦kthen g has a unique fixed point x* in [a,b]. Moreover, for any x in[a,b] the sequence defined byx ₀ =x and x _(n+1) =g(x _(n))converges to x* and for all n

${{x_{n} - x^{*}}} \leq {\frac{k^{n}*{{x_{1} - x_{0}}}}{1 - k}.}$The theorem can be applied CDRF 2 as follows. First, CDRF 2 in apreferred embodiment relates the amount or amounts to be invested acrossthe distribution of states for the CDRF, given a payout distribution, byinverting the expression for the CDRF and solving for the traded amountmatrix A:A=P*Π(A,ƒ)⁻¹  (CDRF 2)CDRF 2 may be rewritten, therefore, in the following form:A=g(A)where g is a continuous and differentiable function. By theaforementioned fixed point theorem, CDRF 2 may be solved by means offixed point iteration if:g′(A)<1i.e., the multivariate function g(A) has a first derivative less than 1.Whether g(A) has a derivative less than 1 with respect to A can beanalyzed as follows. As previously indicated in the specification, forany given trader and any given state i, CDRF 2 contains equations of thefollowing form relating the desired payout p (assumed to be greater than0) to the traded amount α required to generate the desired payout, givena total traded amount already traded for state i of T_(i) (also assumedto be greater than 0) and the total traded amount for all the states ofT:

$\alpha = {\left( \frac{T_{i} + \alpha}{T + \alpha} \right)*p}$ so  that${g(\alpha)} = {\left( \frac{T_{i} + \alpha}{T + \alpha} \right)*p}$Differentiating g(α) with respect to α yields:

${g^{\prime}(\alpha)} = {\left( \frac{T - T_{i}}{T + \alpha} \right)*\frac{p}{T + \alpha}}$Since the DRF Constraint defined previously in the specificationrequires that payout amount p not exceed the total amount traded for allof the states, the following condition holds:

$\frac{p}{T + \alpha} \leq 1$and therefore since

$\left( \frac{T - T_{i}}{T + \alpha} \right) < 1$it is the case that0<g′(α)<1so that the solution to CDRF 2 can be obtained by means of fixed pointiteration as embodied in the computer code listing of Table 1.

1. A method for conducting demand-based trading, comprising the stepsof: establishing a plurality of defined states and a plurality ofpredetermined termination criteria, wherein each of the defined statescorresponds to at least one possible outcome of an event of economicsignificance; accepting, prior to fulfillment of all of thepredetermined termination criteria, an investment of value units by eachof a plurality of traders in at least one of the plurality of definedstates wherein at least one investment of value units is a multi-stateinvestment that designates a set of defined states; and allocating apayout to each investment, responsive to the total number of the valueunits invested in the plurality of defined states, the relative numberof the value units invested in each of the plurality of defined states,and an identification of the defined state that occurred upon thefulfillment of all of the predetermined termination criteria.
 2. Amethod for conducting demand-based trading, comprising the steps of:establishing a plurality of defined states and a plurality ofpredetermined termination criteria, wherein each of the defined statescorresponds to a possible state of a selected financial product wheneach of the predetermined termination criteria is fulfilled; accepting,prior to fulfillment of all of the predetermined termination criteria,an investment of value units by each of a plurality of traders in atleast one of the plurality of defined states wherein at least oneinvestment of value units is a multi-state investment that designates aset of defined states; and allocating a payout to each investment,responsive to the total number of the value units invested in theplurality of defined states, the relative number of the value unitsinvested in each of the plurality of defined states, and anidentification of the defined states that occurred upon the fulfillmentof all of the predetermined termination criteria.
 3. The method forconducting demand-based trading of claims 1 or 2, wherein at least onemulti-state investment designates a set of desired returns responsive tothe designated set of defined states, and wherein the allocating step isfurther responsive to the set of desired returns.
 4. The method forconducting demand-based trading of claim 3, wherein each desired returnof the set of desired returns is responsive to a subset of thedesignated set of defined states.
 5. The method for conductingdemand-based trading of claim 3, wherein the set of desired returnsapproximately corresponds to expected returns from a set of definedstates of a prespecified investment vehicle.
 6. The method forconducting demand-based trading of claim 3, wherein the allocating stepcomprises the step of calculating the required number of value units ofthe multi-state investment that designates a set of desired returns, andthe step of distributing the value units of the multi-state investmentthat designates a set of desired returns to the plurality of definedstates.
 7. The method for conducting demand-based trading of claim 6,wherein the allocating step further comprises the step of solving a setof simultaneous equations that relate traded amounts to unit payouts andpayout distributions; and wherein the calculating step and thedistributing step and responsive to the solving step.
 8. The method forconducting demand-based trading of claim 7, wherein the solving stepcomprises the step of fixed point iteration.
 9. The method forconducting demand-based trading of claim 8, wherein the step of fixedpoint iteration comprises the steps of: selecting an equation of the setof simultaneous equations, the equation having an independent variableand at least one dependent variable; assigning arbitrary values to eachof the dependent variables in the selected equation; calculating thevalue of the independent variable in the selected equation responsive tothe currently assigned values of each of the dependent variables;assigning the calculated value of the independent variable to theindependent variable; designating an equation of the set of simultaneousequations as the selected equation; and sequentially performing thecalculating the value step, the assigning the calculated value step, andthe designating an equation step until the value of each of thevariables converges.